Final answer:
The angle -6 radians places the terminal side in the third quadrant with a reference angle of 0.28 radians when expressed in the positive acute form and rounded to four decimal places.
Explanation:
The question involves determining the quadrant of an angle measured in radians and finding its corresponding reference angle, which is a common task in Mathematics, specifically in the study of trigonometry.
To find the quadrant in which the terminal side of the angle -6 radians lies, we need to recall that one full revolution around the unit circle is 2π or approximately 6.28 radians. Since the given angle is negative, we move in the clockwise direction from the positive x-axis. Dividing -6 by 6.28, we realize that it is just shy of a full negative revolution, thereby placing the terminal side in the third quadrant.
To find the reference angle ¯θ, which is the positive acute angle the terminal side makes with the x-axis, we need to subtract the given angle from one full revolution (if necessary) and find the absolute value. Hence, ¯θ = |2×π - (-6)| = |6.28 - (-6)| = |0.28|, which is the same as 0.28 radians when rounded to four decimal places.
A specific brand of gourmet chocolate candy contains 7.00 gg of dietary fat in each 22.7-gg piece. How many kilograms of dietary fat are in a box containing 4.00 lblb of candy? Express your answer numerically in kilograms.
Answer:
635599482kg
Step-by-step explanation:
A gourmet chocolate candy has 7.00 gg of dietary fat in each 22.7-gg piece
for the 22.7 piece , the gg is 7 x 22.7 = 158.9gg in total
1gg = 2204622.6218488 lbs
158.9gg in total = 158.9 x 2204622.6218488 = 350314534.612
for the box of 4.00 lblb of candy
it is 350314534.612 x 4 = 1401258138.45lb
since our answer is needed in kilograms we convert 1401258138.45lb to kilograms
1lb = 0.453592kg
1401258138.45lb = 1401258138.45lb x 0.453592kg = 635599481.513 ≈ 635599482kg
Scientists at the Hopkins Memorial Forest have been collecting environmental data for more than 100 years. Sulfate content in water samples from Birch Brook is known to be 7.38mg/L with a standard deviation of 1.60mg/L. If 10 students measure the sulfate in their samples to together estimate the mean, what is the probability that they get a sampling error of greater than 1.0 mg/L? What assumptions do we need to make to calculate that value?
Answer:
The probability that they get a sampling error of greater than 1.0 mg/L is 0.04.
To calculate this value, I assume that the samples are randomly collected
Step-by-step explanation:
Sampling error can be calculated using the formula
[tex]\frac{t*s}{\sqrt{n}}[/tex] where
t is the statistic of the probability getting the standard errors is the standard deviation (1.60mg/L)n is the sample size (10)For the sampling error of 1.0mg/L we have
[tex]1=\frac{t*1.60}{\sqrt{10}}[/tex]
Solving for t we have t≈1.976
Then the probability that they get a sampling error of greater than 1.0 mg/L is
P(t>1.976) ≈ 0.04.
In other words, we are 96% confident that the sampling error is within 1.0 mg/L.
To calculate this value, I assume that the samples are randomly collected.
55. If a test statistic falls in the critical region the null hypothesis is _______________. a) Rejected b) Not Rejected c) It depends d) accepted
Answer:
Option a) Rejected
Step-by-step explanation:
We define critical region as:
It is also known as the rejection region.It is the region where the null hypothesis is rejected.Thus, if our calculated test statistic lies in the critical region, we fail to accept the null hypothesis and reject it.For right tailed test: The critical region is the values greater than the critical value.For left tailed test: The critical region is the values less than the critical value.The critical region for two tailed test is the value apart from the range of critical values.Thus,
If a test statistic falls in the critical region the null hypothesis is rejected.
After having a sonogram, a pregnant woman learns that she will have twins. What is the probability that she will have identical twins?
Answer:
[tex]\frac{1}{7}[/tex]
Step-by-step explanation:
The table containing information about the sexes of twins and and the type of twins are given in the table below with corresponding weightage.
Now, since we know that she is expecting twins, the probability of the event
[tex]A - '\text{the woman will have identical twins }'[/tex]
is calculated as follows
[tex]Pr(A) = \dfrac{\text{total number of boy/boy, boy/girl, girl/boy and girl/girl identical twins}}{\text{total number of both identical and frathernal twins}}[/tex]
By using the informations from the given table, we obtain
[tex]Pr(A) = \frac{2 + 0 + 0 +2}{28} = \frac{4}{28} = \frac{1}{7}[/tex]
A boiler has five identical relief valves. The probability that any particular valve will open on demand is 0.93. Assume independent operation of the valves. Calculate P(at least one valve opens). (Round your answer to eight decimal places.)
Answer:
There is a 99.99998% probability that at least one valve opens.
Step-by-step explanation:
For each valve there are only two possible outcomes. Either it opens on demand, or it does not. This means that we use the binomial probability distribution to solve this problem.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
[tex]n = 5, p = 0.93[/tex]
Calculate P(at least one valve opens).
This is [tex]P(X \geq 1)[/tex]
Either no valves open, or at least one does. The sum of the probabilities of these events is decimal 1. So:
[tex]P(X = 0) + P(X \geq 1) = 1[/tex]
[tex]P(X \geq 1) = 1 - P(X = 0)[/tex]
So
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{5,0}.(0.93)^{0}.(0.07)^{5} = 0.0000016807[/tex]
Finally
[tex]P(X \geq 1) = 1 - P(X = 0) = 1 - 0.0000016807 = 0.9999983193[/tex]
There is a 99.99998% probability that at least one valve opens.
Answer:
0.930000
Step-by-step explanation:
An athlete ran 200 meters in 19.19 seconds. Suppose that he ran the first half of that race (around a curve) in 10.75 seconds. How long did it take him to run the second half of the race (on a straight track)?
The time needed to run the second half of the race is 8.44 s
Step-by-step explanation:
The total distance covered by the athlete during the race is
d = 200 m
And the total time taken to cover this distance is
T = 19.19 s
We also know that the time the athlete needs to cover the first half of the race is
[tex]t_1 = 10.75 s[/tex]
Also, we know that
[tex]T=t_1 + t_2[/tex]
where [tex]t_2[/tex] is the time the athlete takes to cover the second half of the race.
Re-arranging this equation and susbtituting the values, we find the value of t2:
[tex]t_2 = T-t_1 = 19.19-10.75=8.44 s[/tex]
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Determine the decision criterion for rejecting the null hypothesis in the given hypothesis test. Describe the values of the test statistic that would result in rejection of the null hypothesis. A cereal company claims that the mean weight of the cereal in its packets is at least 14 oz. You wish to test this claim at the 0.02 level of significance. The mean weight for the random sample of 45 cerial packets is 13.8 oz with a standard deviation of 0.3 oz. What criterion would be used for rejecting the null hypothesis, that oz ? Reject H0 if test statistic < 2.05. Reject H0 if test statistic > -2.05. Reject H0 if test statistic < -2.05. Reject H0 if test statistic < -2.33.
Answer:
We can assume that the sample size is large enough to use the z distribution as an approximation of the t distribution
Now we need to find on the z distribution a value that accumulates 0.02 of the area on the left and this value is [tex]z_{crit}=-2.05[/tex]
We can use the following excel code to verify it: "=NORM.INV(0.02,0,1)"Reject H0 if test statistic < -2.05
And for our case our calculated value <-2.05 so we have enough evidence to reject the null hypothesis at 2% of significance.
Step-by-step explanation:
Data given and notation
[tex]\bar X=13.8[/tex] represent the sample mea n
[tex]s=0.3[/tex] represent the sample standard deviation
[tex]n=45[/tex] sample size
[tex]\mu_o =68[/tex] represent the value that we want to test
[tex]\alpha=0.02[/tex] represent the significance level for the hypothesis test.
t would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value for the test (variable of interest)
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the mean is at least 14 oz, the system of hypothesis would be:
Null hypothesis:[tex]\mu \geq 14[/tex]
Alternative hypothesis:[tex]\mu < 14[/tex]
If we analyze the size for the sample is > 30 but we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:
[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)
t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]t=\frac{13.8-14}{\frac{0.3}{\sqrt{45}}}=-4.47[/tex]
Critical region
We can assume that the sample size is large enough to use the z distribution as an approximation of the t distribution
Now we need to find on the z distribution a value that accumulates 0.02 of the area on the left and this value is [tex]z_{crit}=-2.05[/tex]
We can use the following excel code to verify it: "=NORM.INV(0.02,0,1)"
So then the correct answer for this case would be:
Reject H0 if test statistic < -2.05
And for our case our calculated value <-2.05 so we have enough evidence to reject the null hypothesis at 2% of significance.
The decision criterion to reject the null hypothesis that the mean weight is at least 14 oz at the 0.02 level of significance is to reject H0 if the test statistic is less than -2.33, as derived from the standard normal z-table.
Explanation:To determine the criterion for rejecting the null hypothesis in a hypothesis test like the one presented - where a cereal company claims that the mean weight of the cereal in its packets is at least 14 oz - we use the level of significance (alpha). Since the level of significance in this problem is 0.02, we refer to the standard normal z-table to find the critical z-value that corresponds to this significance level for a one-tailed test, which in this case is a left-tailed test because we are checking if the mean weight is less than the claim (14 oz).
Using the z-table, we find that the critical z-value for an alpha of 0.02 in a one-tailed test is approximately -2.33. If our calculated test statistic is less than this critical value, we would reject the null hypothesis in favor of the alternative. Therefore, the decision criterion is: Reject H0 if test statistic < -2.33.
Since the given options are 'Reject H0 if test statistic < 2.05', 'Reject H0 if test statistic > -2.05', 'Reject H0 if test statistic < -2.05', and 'Reject H0 if test statistic < -2.33', the correct criterion for this test, considering the level of significance is 0.02, is 'Reject H0 if test statistic < -2.33'.
Suppose that there are 2 million inhabitants of a country in which 1 the mean gross (before tax) income is $20,000 per year. Suppose that the mean individual pays taxes on $3000 per year. What is the total disposable income (gross income minus taxes) per year for the country as a whole?
Answer:
$34 billion
Step-by-step explanation:
Data provided in the question:
Total number of inhabitants of a country = 2 million
1 mean gross income = $20,000 per year
Tax paid by mean individual = $3000 per year
Now,
Mean disposable income
= Mean gross income - Mean tax
= $20,000 - $3,000
= $17,000 per year
Therefore,
Total disposable income per year
= Mean disposable income × Total number of inhabitants of a country
= $17,000 × 2 million
= $34 billion
Evaluate the given expression and express the result using the usual format for writing numbers (instead of scientific notation). 29 Upper C 329C3 29 Upper C 329C3equals=nothing
Answer:
29C3 = 3654
Step-by-step explanation:
What is applied here is combination that n - combination of r which is a subscript and the n - is a superscript.
nCr = nPr/r!where nPr = n!/(n-r)!nCr = n!/(n-r)! r!And we are told to find 29C3, apply the formula and your results will be out.Attached is the detailed explanation.
Find the domain of each function. (Enter your answer using interval notation.)(a) f(x) = 81-e^x^2/1-e^81-x^2 (b) f(x) = 7 + x/e^cos x
Answer:
a) (-∞,-9)∪(-9,9)∪(9,+∞)
b) (-∞,+∞)
Step-by-step explanation:
The domain of a real function is the largest subset og the real line in which it is defined.
a) The function [tex]f(x)=\frac{81-e^{x^2}}{1-e^{81-x^2}}[/tex] is defined for all values of x in which the denominator is not zero. The denominator is zero if [tex]0=1-e^{81-x^2}[/tex], that is, [tex]e^{81-x^2}=1[/tex]. The exponential function is equal to one only if the exponent is equal to zero, then the values of x that nullify the denominator satisfy [tex]81-x^2=0[/tex], thus x=9 or x=-9. Then the domain of f is the set of all real numbers such that x≠9 and x≠-9, which is the set (-∞,-9)∪(-9,9)∪(9,+∞).
b) In this case the denominator is [tex]e^{\cos x}[/tex] which is always positive. Thus the denominator is not zero for all real x, then the fomain is the real line, which is the interval (-∞,+∞).
Final answer:
The domain of the function f(x) = (81-e^x^2)/(1-e^81-x^2) is (-∞, -9) ∪ (-9, 9) ∪ (9, ∞). The domain of the function f(x) = 7 + x/e^cos(x) is (-∞, ∞).
Explanation:
(a) To find the domain of the function f(x) = (81-e^x^2)/(1-e^81-x^2), we need to determine the values of x for which the function is defined. The function is defined as long as the denominator is not zero. So, we need to solve the equation 1 - e^(81-x^2) = 0 to find where the denominator equals zero. Simplifying this equation, we get e^(81-x^2) = 1. Taking the natural logarithm of both sides, we have 81-x^2 = 0. Therefore, x^2 = 81, and x = ±9. The domain of the function is (-∞, -9) ∪ (-9, 9) ∪ (9, ∞).
(b) To find the domain of the function f(x) = 7 + x/e^cos(x), we need to determine the values of x for which the function is defined. The function is defined as long as the denominator e^cos(x) is not zero. Since the exponential function is always positive, we know that e^cos(x) will never be zero. Therefore, the domain of the function is the set of all real numbers, (-∞, ∞).
A van starts off 191 miles directly north from the city of Morristown. It travels due east at a speed of 25 miles per hour. After travelling 119 miles, how fast is the distance between the van and Morristown changing?
Answer:
Distance between the van and Morristown is changing at the rate of 13.22 miles per hour.
Step-by-step explanation:
From the figure attached,
Van starts from C (City of Morristown), reaches the point A 191 miles due North and then it travels with a speed of 25 miles per hour due East from A towards B.
We have to calculate the rate of change of distance BC, when the van reaches point B which is 119 miles away from A.
By Pythagoras theorem in the triangle ABC,
[tex]BC^{2}=AB^{2}+AC^{2}[/tex]
Distance AC is constant equal to 191 mi.
By differentiating the equation with respect to time 't'
[tex]2BC.\frac{d(BC)}{dt}=0+2AB.\frac{d(AB)}{dt}[/tex]
[tex]BC.\frac{d(BC)}{dt}=AB.\frac{d(AB)}{dt}[/tex]
Since BC² = (119)²+ (191)²
BC = √50642 = 225.04 miles
From the differential equation,
[tex](225.03).\frac{d(BC)}{dt}=119\times 25[/tex] [Since [tex]\frac{d(AB)}{dt}=25[/tex] miles per hour]
[tex]\frac{d(BC)}{dt}=13.22[/tex] miles per hour
Therefore, distance between the van and Morristown is changing at the rate of 13.22 miles per hour.
A professor gives a statistics exam. The exam has 100 possible points. The scores for the students in the second classroom are as follows: 88 84 84 52 100 84 92 84 88 Calculate the sample size, n, and the sample mean, M.
Answer: Sample size n is 9.
The sample mean M is 84.
Step-by-step explanation:
Given : A professor gives a statistics exam.
The exam has 100 possible points.
The scores for the students in the second classroom are as follows:
88 84 84 52 100 84 92 84 88
Since , the number of scores of different students= 9
∴ Sample size is n= 9.
Now , Sample mean = [tex]\dfrac{\text{Sum of all data values}}{\text{Sample size}}[/tex]
[tex]=\dfrac{88+ 84 +84 +52 +100+ 84+ 92+ 84+ 88 }{9}\\\\=\dfrac{756}{9} =84[/tex]
Hence, the sample mean M is 84.
We have that the sample size, n, and the sample mean, M. is mathematically given as
n=9M_n=88From the question we are told
A professor gives a statistics exam.
The exam has 100 possible points. The scores for the students in the second classroom are as follows: 88 84 84 52 100 84 92 84 88 Calculate the sample size, n, and the sample mean, M.MeanGenerally the equation for the Mean is mathematically given as
[tex]M=\sum/n\\\\\Therefore\\\\M=\frac{88 +84 +84+ 52+ 100+ 84+ 92+ 84+ 88 }{9}\\\\M=84[/tex]
Therefore
New mean
M_n=84+4
M_n=88
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An automobile manufacturer claims that its van has a 39.0 miles/gallon (MPG) rating. An independent testing firm has been contracted to test the MPG for this van since it is believed that the van has an incorrect manufacturer's MPG rating. After testing 120 vans, they found a mean MPG of 38.8. Assume the standard deviation is known to be 2.2. A level of significance of 0.02 will be used. Find the value of the test statistic. Round your answer to 2 decimal places. Enter the value of the test statistic.
Final answer:
To evaluate the manufacturer's claim, a one-sample t-test can be conducted. The test statistic can be calculated using the given information.
Explanation:
To conduct a hypothesis test to evaluate the manufacturer's claim, we can use a one-sample t-test. The null hypothesis (H0) is that the mean MPG is equal to 39.0, and the alternative hypothesis (Ha) is that the mean MPG is not equal to 39.0. We can calculate the test statistic using the formula:
test statistic = (sample mean - population mean) / (standard deviation / sqrt(sample size))
Using the given values:
sample mean = 38.8
population mean = 39.0
standard deviation = 2.2
sample size = 120
Substituting these values, the test statistic is:
test statistic = (38.8 - 39.0) / (2.2 / sqrt(120))
A die is thrown twice. Let X1 and X2 denote the outcomes, and define random variable X to be the minimum of X1 and X2. Determine the distribution of X.
Answer:
Step-by-step explanation:
Given that a die is thrown twice.
X1, X2 are the outcomes in 2 throws
X1 = minimum of two
X1 can be either 1 or 2...6
Total outcomes are 36.
For x1 =1, fav ourable outcomes are (1,1) (1,2)...(1,6) (6,1)...(2/,1)= 11
P(X1=1) = [tex]\frac{11}{36}[/tex]\
P(X1=2) =Prob for one die showing two and other die showing 2 to 6
= [tex]\frac{9}{36}[/tex]
P(x1=3) = Prob for one die showing three and other die showing 3 to 6
=[tex]\frac{7}{36}[/tex]
thus we find that probability is reducing by 2 in the numerator
P(x1=4) = 5/36 followed by 3/36 for 5 and 1/36 for 6
Final answer:
The random variable X represents the minimum outcome when rolling a die twice. The values of X range from 1 to 6 and its distribution is given.
Explanation:
The random variable X represents the minimum outcome when rolling a die twice. In other words, X is the smaller of the two outcomes.
X can take on values from 1 to 6 since the outcomes of rolling a die are integers between 1 and 6. The minimum value of X is 1, which occurs when both dice show a 1.
The distribution of X is as follows:
p(X = 1) = 1/36p(X = 2) = 3/36p(X = 3) = 5/36p(X = 4) = 7/36p(X = 5) = 9/36p(X = 6) = 11/36What dose the equation P =?
I think $8.95
Answer: Yes, you are correct it's $8.95 because like they stated in the word problem p= the price of each bowl and like they said in the word problem they charge $8.95 a bowl. So, $8.95 is the answer.
Step-by-step explanation:
Of all the numbers whose difference is 36, find the two that have the minimum product
Answer:
54,18
Step-by-step explanation:
Let one of the numbers be x. The other number be represented as 36-x
(x-(36-x ) = 36
open brackets
x - 36 + x = 36
x + x = 36+ 36 = 72
2x = 72
x = 72/2 = 36
x = 36
The product can then be represented as y = x(36-x) or y=36x-x²
The maximum or minimum is always on the axis of symmetry which has the formula x=-b/2a.
In our case, the axis of symmetry is -36/-2, so x=18.
If one number is 18 and the 2 numbers differentiate by 36, the other number is 18 + 36 = 54
So the 2 numbers are 18 and 54 and the minimum product is 972
The two numbers are 18 and 54 and the minimum product is 972
What is the fundamental principle of multiplication?If an event can occur in m different ways and if following it, a second event can occur in n different ways, then the two events in succession can occur in m × n different ways.
Let suppose one of the numbers be x and the other number be represented as 36-x
Then the equation would be;
(x-(36-x ) = 36
Now open brackets;
x - 36 + x = 36
x + x = 36+ 36 = 72
2x = 72
x = 72/2 = 36
x = 36
Therefore, The product can be represented as y = x(36-x) or y=36x-x²
Here maximum or minimum is always on the axis of symmetry which is the formula x=-b/2a.
In our case; the axis of symmetry is -36/-2,
Then x=18.
If one number is 18 and the 2 numbers differentiate by 36 then the other number must be 18 + 36 = 54
Hence the two numbers are 18 and 54 and the minimum product is 972
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In a sample of three people, the first person has a score 5 points above the mean, and the second person has a score 3 points above the mean.
How does the third persons score compare to the mean?
Answer:
8 points below the mean.
Step-by-step explanation:
Let 'X' be the score of the third person, and let 'M' be the mean score of the sample.
If the other two people scored 5 and 3 above the mean, in order to maintain consistency, the following expression must be true:
[tex]M+5+(M+3)+X =3M\\X=3M-2M-8\\X=M-8[/tex]
Therefore, the third person has a score 8 points below the mean.
Given that the other two people's scores are collectively 8 points above the mean, the third person's score must be 8 points below the mean to balance out the total deviation.
Explanation:This question is about determining the score of the third person in a sample relative to the mean of the sample. We know that the total deviation from the mean must equal zero because the mean is the average of all the scores. Given the first person has a score 5 points above the mean, and the second person has a score 3 points above the mean, the total deviation above the mean is (5+3)=8 points. For the score deviations to balance out to zero, the third person's score must be 8 points below the mean.
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We have a bag filled with 201 marbles, of which 100 of them are blue and 101 of them are red. Every turn, we remove 2 marbles from the bag. If the two marbles are of the same color, we remove the two marbles but add a blue marble into the bag. If the two marbles are of different colors, we remove the two marbles and add a red marble into the bag. What is the color of the last marble in the bag?
Answer:
Red Marble is the color in the bag
Step-by-step explanation:
Total number of marbles ; 201
number of red marbles ; 101
number of blue marbles ; 101
Every turn, 2 marbles of the same color are removed, and add a blue marbleif two marbles are different color are removed , and add a red marbleFrom the analysis of the question; there are three possibilities;first possibility ; From what is said as the condition, it can be inferred that the number of blue marbles is reduced by a factor of 1 , while the number of red marbles remains constant. this shows that the two marbles of the same color that was removed is most likely BLUE.
Second possibility ; Two red marbles are added (of the same color) and add a blue marbles, implies the the red marbles reduce by a factor of 2 while the blue marbles increase by a factor of 1. this is also possible.
Third possibility ; Two marbles of different color implies 1R and 1B , and then a red marble is added. this shows that the number of red marble still remains constant ( 1R removed, 1R added), while the number of blue reduce by 1 ( remove 1B, and not replaced back).
From the various possibilities, it can be inferred that the number of red marbles is either increasing by a factor of 2 or remains constant. this further shows that in all three , 1R marble will always remain in the bag no matter the possibility as such in all three scenerios, it further shows that if there is only a marble remaining, it would have been the RED MARBLE WHICH IS AUTOMATICALLY THE LAST MARBLE from the analysis done above.
"If a ball is thrown into the air with a velocity of 44 ft/s, its height in feet t seconds later is given by y = 44t − 16t2. (a) Find the average velocity for the time period beginning when t = 2 and lasting for each of the following. (i) 0.5 seconds ft/s (ii) 0.1 seconds ft/s (iii) 0.05 seconds ft/s (iv) 0.01 seconds ft/s (b) Estimate the instantaneous velocity when t = 2. ft/s"(c) Find the average velocity for the time period beginning whent = 2 and lasting 0.05 second. ft/s(d) Find the average velocity for the time period beginning whent = 2 and lasting 0.01 second. ft/s(e) Estimate the instantaneous velocity when t = 2.
Answer:
i) -28 ft/s
ii) -21.6 ft/s
iii) -20.8 ft/s
iv) -20.16 ft/s
b) -20 ft/s
Step-by-step explanation:
if v represents vertical velocity then if the height y is
y = 44*t − 16*t²
and the instant velocity v is the derivative with respect to the time
v= dy/dt = 44*(1) - 16*(2*t) = 44- 32*t
while the average velocity is va= (y-y₀)/(t-t₀)
where t₀ = 2 and y₀=y(t₀) = 44*2 − 16*2² = 24
then
va= (y-y₀)/(t-t₀) = (44*t − 16*t² - 24)/(t-2)
for
i) t= 0.5s + 2 s= 2.5 s
va = (44* 2.5 − 16* 2.5 ² - 24)/( 2.5 -2) = -28 ft/s
ii) t= 0.1s + 2 s= 2.1 s
va = (44* 2.1 − 16* 2.1 ² - 24)/( 2.1 -2) = -21.6 ft/s
iii) t= 0.05 s + 2 s= 2.05 s
va = (44* 2.05 − 16* 2.05 ² - 24)/( 2.05 -2) = -20.8 ft/s
iv) t= 0.01s + 2 s= 2.01 s
va = (44* 2.01 − 16* 2.01 ² - 24)/( 2.01 -2) = -20.16 ft/s
b) the instantaneous velocity when t=2
v (t=2) = 44- 32*(2) = -20 ft/s
Answer:
i) The first -28 ft/s
ii) Then second -21.6 ft/s
iii) Thirds one is -20.8 ft/s
iv) Then -20.16 ft/s
b) -20 ft/s
Step-by-step explanation:When if v represents vertical velocity then if the height y isthen y = 44*t − 16*t²After that and the instant velocity v is the derivative with respect to the timeNow v= dy/dt = 44*(1) - 16*(2*t) = 44- 32*tAlso that while the average velocity is va= (y-y₀)/(t-t₀)After that where t₀ = 2 and y₀=y(t₀) = 44*2 − 16*2² = 24then va= (y-y₀)/(t-t₀) = (44*t − 16*t² - 24)/(t-2)for (i) t= 0.5s + 2 s= 2.5 sNow va = (44* 2.5 − 16* 2.5 ² - 24)/( 2.5 -2) = -28 ft/sNext (ii) t= 0.1s + 2 s= 2.1 sAfter that va = (44* 2.1 − 16* 2.1 ² - 24)/( 2.1 -2) = -21.6 ft/sNow (iii) t= 0.05 s + 2 s= 2.05 sThen va = (44* 2.05 − 16* 2.05 ² - 24)/( 2.05 -2) = -20.8 ft/siv) t= 0.01s + 2 s= 2.01 sAfter that va = (44* 2.01 − 16* 2.01 ² - 24)/( 2.01 -2) = -20.16 ft/sWhen b) the instantaneous velocity then t=2Now v (t=2) = 44- 32*(2) = -20 ft/sLearn more about:
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A poll asked people in a certain country, "Thinking about your finances today, would you say that it is now a good time or a bad time to take a lengthy vacation?
The choices were: "Good time" or "Bad time."
a. What kind of variable is the response?
Answer:
Qualitative variable
Step-by-step explanation:
The response variable is qualitative because it can be categorize as "Good time" or "Bad time". The time is also rated as "Good" or "Bad" for taking a lengthy vacation and so it involves ordinal scale of measurement.
We can say that our response variable is qualitative and ordinal. In short response variable is of qualitative type of variable and the scale of measurement it involves is ordinal scale.
What function do you know from calculus is such that its first derivative is itself? Its first derivative is a constant multiple of itself? Write each answer in the form of a first-order differential equation with a solution.
Answer:
First Part:
exponential function [tex]e^t[/tex] is the one whose first order derivative is the function itself.
Second Part:
[tex]y=ce^{At}\\y'=Ay[/tex]
Third Part:
[tex]y=ce^t\\y'=ce^t\\y'=y[/tex]
Step-by-step explanation:
First Part:
In calculus exponential function [tex]e^t[/tex] is the one whose first order derivative is the function itself.
Where:
t is independent variable.
Derivative is represented as:
[tex]y=ce^t\\y'=\frac{d(ce^t)}{dt} \\y'=ce^t\\y'=y[/tex]
Where:
c is any number.
Second Part:
Consider the constant A.
The function will become:
[tex]y=ce^{At}\\y'=\frac{d(ce^{At})}{dt} \\y'=cAe^{At}\\y'=Ay[/tex]
Third Part:
Derivative is represented as:
[tex]y=ce^t\\y'=\frac{d(ce^t)}{dt} \\y'=ce^t\\y'=y[/tex]
Where:
c is any number.
Final answer:
The differential equation is [tex]\( \frac{df}{dx} = Cf(x) \),[/tex] with the solution[tex]\( f(x) = Ce^x \), where \( C \)[/tex] is a constant.
Explanation:
The function that satisfies both conditions is [tex]\( f(x) = Ce^x \), where \( C \)[/tex] a constant. Its first derivative is[tex]\( f'(x) = Ce^x \),[/tex]which is a constant multiple of itself. This can be expressed as a first-order differential equation:
[tex]\[ \frac{df}{dx} = Cf(x) \][/tex]
with the solution:
[tex]\( f(x) = Ce^x \), where \( C \)[/tex]
This differential equation represents exponential growth or decay, depending on the sign of [tex]\( C \). For \( C > 0 \)[/tex], the function exponentially grows as [tex]\( x \) increases, while for \( C < 0 \)[/tex]it exponentially decays. The solution captures phenomena like population growth, radioactive decay, or compound interest, making it a fundamental concept in calculus and applied mathematics.
Find the exact value of cos theta, given that sin thetaequalsStartFraction 15 Over 17 EndFraction and theta is in quadrant II. Rationalize denominators when applicable.
Answer:
[tex] cos \theta = -\frac{8}{17}[/tex]
Step-by-step explanation:
For this case we know that:
[tex] sin \theta = \frac{15}{17}[/tex]
And we want to find the value for [tex] cos \theta[/tex], so then we can use the following basic identity:
[tex] cos^2 \theta + sin^2 \theta =1 [/tex]
And if we solve for [tex] cos \theta [/tex] we got:
[tex] cos^2 \theta = 1- sin^2 \theta[/tex]
[tex] cos \theta =\pm \sqrt{1-sin^2 \theta}[/tex]
And if we replace the value given we got:
[tex] cos \theta =\pm \sqrt{1- (\frac{15}{17})^2}=\sqrt{\frac{64}{289}}=\frac{\sqrt{64}}{\sqrt{289}}=\frac{8}{17}[/tex]
For our case we know that the angle is on the II quadrant, and on this quadrant we know that the sine is positive but the cosine is negative so then the correct answer for this case would be:
[tex] cos \theta = -\frac{8}{17}[/tex]
Answer:
It is D
Step-by-step explanation:
EDGE 2021
In each of Problems 5 through 10, verify that each given function is a solution of the differential equation.
y''-y=0, y1(t)=e^t , y2(t)=cosht
Answer:
For First Solution: [tex]y_1(t)=e^t[/tex]
[tex]y_1(t)=e^t[/tex] is the solution of equation y''-y=0.
For 2nd Solution:[tex]y_2(t)=cosht[/tex]
[tex]y_2(t)=cosht[/tex] is the solution of equation y''-y=0.
Step-by-step explanation:
For First Solution: [tex]y_1(t)=e^t[/tex]
In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.
[tex]y_1(t)=e^t[/tex]
First order derivative:
[tex]y'_1(t)=e^t[/tex]
2nd order Derivative:
[tex]y''_1(t)=e^t[/tex]
Put Them in equation y''-y=0
e^t-e^t=0
0=0
Hence [tex]y_1(t)=e^t[/tex] is the solution of equation y''-y=0.
For 2nd Solution:
[tex]y_2(t)=cosht[/tex]
In order to prove whether it is a solution or not we have to put it into the equation and check. For this we have to take derivatives.
[tex]y_2(t)=cosht[/tex]
First order derivative:
[tex]y'_2(t)=sinht[/tex]
2nd order Derivative:
[tex]y''_2(t)=cosht[/tex]
Put Them in equation y''-y=0
cosht-cosht=0
0=0
Hence [tex]y_2(t)=cosht[/tex] is the solution of equation y''-y=0.
The functions [tex]y(t)=e^t[/tex] and [tex]y(t)=cosht[/tex] are solutions of given differential equation.
We have to prove that given function is a solution of the differential equation.
Given function, [tex]y(t)=e^t[/tex]
Given differential equation are, [tex]y''-y=0[/tex]
So that, [tex]y'(t)=e^t , y''(t)=e^t[/tex]
Substitute values in given differential equation.
[tex]e^t - e^t=0[/tex]
Thus, function [tex]y(t)=e^t[/tex] is solution of given differential equation.
Another function is, [tex]y(t)=cosht[/tex]
So that, [tex]y'(t)=sinht , y''(t)=cosht[/tex]
Substitute values in given differential equation.
[tex]cosht-cosht=0[/tex]
Thus, function [tex]y(t)=cosht[/tex] is solution of given differential equation.
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Fertilizer is sold in 100 pound bags labelled with the amount of nitrogen ( N), phosphoric acid ( P2O5), and potash ( K2O) present. The mixture of these nutrients varies from one type of fertilizer to the next. For example, a 100 pound bag of Vigoro Ultra Turf fertilizer contains 29 pounds of nitrogen, 3 pounds of phosphoric acid, and 4 pounds of potash. Another type of fertilizer, Parkerâ's Premium Starter, has 18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash per 100 pounds. Determine the amount of each type required to yield a mixture containing the 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash.
Answer:
4 pounds of of Vigoro Ultra Turf fertilizer and 1 pound of Parkera's Premium Starter will yield the required mixture.
Step-by-step explanation:
In bag of Vigoro Ultra Turf fertilizer:
29 pounds of nitrogen, 3 pounds of phosphoric acid, and 4 pounds of potash
In bag of Parkera's Premium Starter:
18 pounds of nitrogen, 25 pounds of phosphoric acid, and 6 pounds of potash
Let the mass of Vigoro Ultra Turf fertilizer = x
Amount of nitrogen in x amount = 29 x
Amount of phosphoric acid in x amount = 3x
Let the mass of Parkera's Premium Starter= y
Amount of nitrogen in y amount = 18 y
Amount of phosphoric acid in y amount = 25 y
Amount of nitrogen in desired mixture formed by combing x and y fertilizers bag : 134 pounds
29 x + 18 y = 134 ..[1]
Amount of phosphoric acid in desired mixture formed by combing x and y fertilizers bag : 37 pounds
3x + 25 y = 37 ..[2]
Solving [1] anf [2] by substitution method :
[tex]x=\frac{134-18y}{29}[/tex]
[tex]3\times \frac{134-18y}{29}+25y=37[/tex]
y = 1
[tex]x=\frac{134-18y}{29}=4[/tex]
4 pounds of of Vigoro Ultra Turf fertilizer and 1 pound of Parkera's Premium Starter will yield the required mixture.
To create a mixture containing 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash, 400 pounds of Vigoro Ultra Turf and 100 pounds of Parker's Premium Starter are required, based on solving a system of linear equations.
We are tasked with finding the amount of two different fertilizers needed to obtain a mixture with 134 pounds of nitrogen, 37 pounds of phosphoric acid, and 22 pounds of potash. The two fertilizers are Vigoro Ultra Turf and Parkerâ's Premium Starter. We'll use a system of linear equations to solve this problem.
Let V represent the amount (in pounds) of Vigoro Ultra Turf and P represent the amount (in pounds) of Parkerâ's Premium Starter. The equations based on the given information are:
0.29V + 0.18P = 134 (nitrogen content)0.03V + 0.25P = 37 (phosphoric acid content)0.04V + 0.06P = 22 (potash content)Now we solve these equations simultaneously to determine V and P. After solving, we find that V = 400 and P = 100. Thus, we need 400 pounds of Vigoro Ultra Turf and 100 pounds of Parkerâ's Premium Starter to obtain the desired mixture.
12 ounces is roughly the same
Answer:
Roughly the same as what?
Step-by-step explanation:
Answer:
A. 400 GRAMS
B. 120 GRAMS
C. 356 GRAMS
D. 340 GRAMS
PENN FOSTER
ANSWER IS D 340 GRAMS
Step-by-step explanation:1 ounce = 28.3495 12 ounces = 28.3495 * 12 = 340.194 grams
P (A or B ). P(A or B) indicates _______. Choose the correct answer below. A. the probability that A and B both occur in the same trial. B. the probability that in a single trial, event A occurs, event B occurs, or they both occur. C. the probability that event A occurs in one trial followed by event B in another trial. D. the probability that event A or event B does not occur in a single trial.
Answer:
Option B is right
Step-by-step explanation:
P(A or B)
In set theory if A and B are two sets we have either A occurs or B occurs or both as
P(A or B) = P(AUB)
This implies that A occurs or B occurs
Probability is calculated as P(A)+P(B)-P(AB)
P(AB) here represents both occurring and this is subtracted as this was added two times
Option A is wrong because this is AB
Option C is wrong because it is not necessary A occurs first
D is wrong because this is A'UB' where A' represents the complement of A
So correct answer out of four options is
B. the probability that in a single trial, event A occurs, event B occurs, or they both occur.
The notation P(A or B) in probability specifies the probability that in a single trial, either event A happens, event B happens, or they both happen.
Explanation:The notation P(A or B) in probability theory denotes the probability that event A occurs, event B occurs, or both events occur in a single trial. If we look at the four options provided, option B fits this definition. Hence, P(A or B) indicates the probability that in a single trial, event A occurs, event B occurs, or they both occur. It does not imply that both events A and B have to occur concurrently in the same trial (option A), nor that event A has to occur in one trial followed by event B in another trial (option C), and certainly not the probability that event A or event B does not occur in a single trial (option D).
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Geno wants to purchase gym membership. He has no more than y dollars to spend. Total Fitness charges an initial fee of $100 plus $30 per month. Gymania charges initial fee of $25 plus $50 per month. Write a system of equations that can be used to determine which company offers the better deal.
Answer:
Gymania is a better deal if the membership is for 3 months and below.
Total Fitness is a better if the membership is for 4 months and above.
Step-by-step explanation:
Let the number of months be 'x'.
Given:
Money Geno has = 'y' dollars.
Total Fitness charges:
Monthly fee = $30
Initial fee = $100
Gymania charges:
Monthly fee = $50
Initial fee = $25
Total charges is equal to the sum of initial fee and monthly fee multiplied by number of months.
So, for 'x' months, monthly fee charged by Total Fitness = [tex]30x[/tex]
For 'x' months, monthly fee charged by Gymania = [tex]50x[/tex]
Now, total charge by Total Fitness = Initial fee + Fee for 'x' months
Total charge by Total Fitness = [tex]100+30x[/tex]
Now, total charge by Gymania = Initial fee + Fee for 'x' months
Total charge by Gymania = [tex]25+50x[/tex]
Now, Geno has only 'y' dollars to spend. So, 'y' must be less than or equal to the total charge.
Therefore, the total charge for each membership is:
[tex]y=30x+100\\\\y= 50x+25[/tex]
Now, we graph both the equations. The graph is shown below.
From the graph, it is clear that, the total cost for Gymania (blue line) is less than that of Total Fitness (red line) till number of months equals 3.75 or 3 months. After 3.75 months, the graph of Gymania is above Total Fitness. So, if the membership is 4 months or above, then Total Fitness is more efficient.
Therefore, Gymania is a better deal if the membership is for 3 months and below.
Total Fitness is a better if the membership is for 4 months and above.
To determine which company offers the better deal for a gym membership, set up a system of equations for the two companies' costs and compare. Total Fitness charges $30 per month, while Gymania charges $50 per month. The equation is solved to find the break-even point where their costs are equal.
Explanation:To determine which company offers the better deal, we can set up a system of equations based on the given information:
Let x be the total number of months for the gym membership.
Total Fitness charges an initial fee of $100 plus $30 per month, so the total cost can be represented by the equation: y = 30x + 100.
Gymania charges an initial fee of $25 plus $50 per month, so the total cost can be represented by the equation: y = 50x + 25.
To compare the two deals, we need to find the values of x where the total cost is the same for both companies. We can set up the following equation:
30x + 100 = 50x + 25.
Simplifying, we get:
20x = 75.
Dividing both sides by 20, we find that x = 3.75.
Since x represents the number of months, it cannot be a decimal, so we round up to the nearest whole number. Therefore, Geno should join Total Fitness if he plans to have the membership for 4 or more months, and Gymania if he plans to have the membership for 3 or fewer months.
The equation r(t)=cos(5t)i + sin(5t)j, 0t≥0 describes the motion of a particle moving along the unit circle. Answer the following questions about the behavior of the particle. a. Does the particle have constant speed? If so, what is its constant speed? b. Is the particle's acceleration vector always orthogonal to its velocity vector? c. Does the particle move clockwise or counterclockwise around the circle? d. Does the particle begin at the point left parenthesis 1 comma 0 right parenthesis(1,0)?
Answer:
a) 5 units/s
b) yes
c) counter-clockwise
d) yes
Step-by-step explanation:
part a
r(t) = cos (5t) i + sin (5t)j
v(t) = dr(t) / dt = -5sin(5t) i + 5cos(5t)j
[tex]mag( v(t)) = \sqrt{(-5sin(5t))^2 + (5cos(5t))^2} \\mag( v(t)) = \sqrt{25sin^2(5t) + 25cos^2(5t)} \\ \\mag( v(t)) = \sqrt{25*(sin(5t)^2 + cos(5t)^2)} \\\\mag( v(t)) = \sqrt{25} \\\\mag( v(t)) = 5 units/s[/tex]
Hence, the particle has a constant speed of 5 units/s
part b
a(t) = dv(t) / dt = -25cos(5t) i - 25sin(5t)j
To check orthogonality of two vectors their dot product must be zero
a(t) . v(t) = (-25cos(5t) i - 25sin(5t)j) . (-5sin(5t) i + 5cos(5t)j)
= 125cos(5t)*sin(5t) -125cos(5t)*sin(5t)
= 0
Yes, the particles velocity vector is always orthogonal to acceleration vector.
part c
Use any two values of t and compute results of r(t)
t = 0 , r(0) = 1 i
t = pi/2, r(0) = j
Hence we can see that the particle moves counter-clockwise
part d
Find the value r(t) at t=0
r(0) = cos (0) i + sin (0) j
r(0) = 1 i + 0 j
Yes, the particle starts at point ( 1, 0)
The domain of discourse are the students in a class. Define the predicates: S(x): x studied for the test A(x): x received an A on the test What is the logical expression that is equivalent to?
Answer:
[tex] (\forall x) S(x)\rightarrow A(x) [/tex]
[tex] (\exists x)\neg S(x)\wedge A(x) [/tex]
Step-by-step explanation:
Everyone who studied for the test received an A on the test.
That means- if you studied for the test you will recived an A, and it is hold for everyone, so we will use quantifire [tex]\forall[/tex].
[tex] (\forall x) S(x)\rightarrow A(x) [/tex]
It means: for every student holds- If it is correct that student x studied then the student got an A.
Someone who did not study for the test received an A on the test.
It means that, there is at least one student hwo didn't studie but student got an A. So we have conjuction of two sentences (negation of the S(x) and A(x) for some student- for that we use existential quantifie).
[tex] (\exists x)\neg S(x)\wedge A(x) [/tex]
Final answer:
The logical expression involving predicates S(x) and A(x) can be represented by ∀x(S(x) → A(x)), stating that all students who studied for the test received an A.
Explanation:
The question involving the predicates S(x): x studied for the test and A(x): x received an A on the test revolves around predicate logic, where we aim to understand and analyze the logical relations of sentences with subjects and predicates within a specified domain of discourse.
To define the logical expression that describes a relationship between studying for a test and receiving an A would depend on the specific relationship we want to express. For instance, a possible logical expression could be ∀x(S(x) → A(x)), which translates to 'For all students x in the class, if x studied for the test, then x received an A on the test.'
First-order differential equations need not have unique solutions satisfying a given initial condition.
a)Prove that there are infinitely many different solutions of the differential equations x- = x^1/3 satisfying x(0) = 0.
b)Discuss the corresponding situation that occurs for x- = x/t, x(0) = x_0.
c)Discuss the situation that occurs for x- = x/t^2, x(0) = 0.
Answer:
a) No Unique solution
b) No Unique solution
c) No unique solution
Step-by-step explanation:
The step by step explanation is as shown in the attachment below