Final answer:
The statement forms p → q ∨ r, p ∧ ¬q → r, and p ∧ ¬r → q are shown to be logically equivalent using principles of logical equivalence and contraposition. The original statement 'If n is prime, then n is odd or n is 2' can thus be rephrased in different ways while retaining the same meaning.
Explanation:
Logical Equivalences
To show that the statement forms p → q ∨ r, p ∧ ¬q → r, and p ∧ ¬r → q are all logically equivalent, we need to understand that logical equivalence means that the statements have the same truth value in every possible scenario. The implication p → q is equivalent to p∨ q, as a truth table would confirm. Likewise, by using the principle of contraposition, which states that p → q is equivalent to ¬q → ¬p, we can transform the given implications accordingly.
Considering the sentence "If n is prime, then n is odd or n is 2" and denoting p as "n is prime", q as "n is odd", and r as "n is 2", the original statement can be represented as p → q ∨ r. Using logical equivalences established earlier, two different ways to rewrite this sentence could be:
The logical equivalences show that all these forms express the same condition from different perspectives but with the same meaning.
A function f is described by f(x)=A*exp(kx)+B, where A, B and k are constants. Given f(0)=1, f(1)=2, and that the horizontal asymptote of f is -4, the value of k is
Answer:
k = ln (6/5)
Step-by-step explanation:
for
f(x)=A*exp(kx)+B
since f(0)=1, f(1)=2
f(0)= A*exp(k*0)+B = A+B = 1
f(1) = A*exp(k*1)+B = A*e^k + B = 2
assuming k>0 , the horizontal asymptote H of f(x) is
H= limit f(x) , when x→ (-∞)
when x→ (-∞) , limit f(x) = limit (A*exp(kx)+B) = A* limit [exp(kx)]+B* limit = A*0 + B = B
since
H= B = (-4)
then
A+B = 1 → A=1-B = 1 -(-4) = 5
then
A*e^k + B = 2
5*e^k + (-4) = 2
k = ln (6/5) ,
then our assumption is right and k = ln (6/5)
The value of k is [tex]k=ln(\frac{6}{5} )[/tex].
Given function is,
[tex]f(x)=Ae^{kx} +B[/tex]
Substitute [tex]f(0)=1,f(1)=2[/tex] in above equation.
We get,
[tex]A+B=1\\\\Ae^{k}+B=2[/tex]
Given that horizontal asymptote of f is -4.
[tex]\lim_{x \to -\infty} Ae^{kx}+B=-4\\ \\ B=-4[/tex]
So, [tex]A=1-B=1-(-4)=5[/tex]
Substitute value of A and B.
[tex]5e^{k}-4=2\\ \\e^{k} =\frac{6}{5}\\ \\k=ln(\frac{6}{5} )[/tex]
Hence, the value of k is [tex]k=ln(\frac{6}{5} )[/tex].
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If z=3−5i, find |z|.
Answer:
Step-by-step explanation:
The absolute value of z is the distance between the point graphed from the complex number and the origin on a complex plane. In a complex plane, the x axis is replaced by R, real numbers, and the y axis is replaced by i, the complex part of the complex number. Our real number is positive 3 and the complex number is -5, so we go to the right 3 and then down 5 and make a point. Connect that point to the origin and then connect the point to the x axis at 3 to construct a right triangle that has a base of 3 and a length of -5. To find the distance of the point to the origin is to find the length of the hypotenuse of that right triangle using Pythagorean's Theorem. Therefore:
[tex]|z|=\sqrt{(3)^2+(-5)^2}[/tex] and
[tex]|z|=\sqrt{9+25}[/tex] and
[tex]|z|=\sqrt{34}[/tex]
Is the following variable categorical or quantitative? Collect data from a sample of teenagers with a question that asks ‘‘Do you eat at least five servings a day of fruits and vegetables?""
The survey question 'Do you eat at least five servings a day of fruits and vegetables?' is designed to collect categorical data, as it classifies respondents into groups based on their affirmative or negative answer, rather than providing a numerical value.
Explanation:The question "Do you eat at least five servings a day of fruits and vegetables?" is designed to collect categorical data. This is because the answers to the question will classify respondents into different categories, specifically those who do eat at least five servings of fruits and vegetables per day and those who do not. As such, the data obtained will be qualitative in nature, allowing us to compare and organize individuals based on their dietary habits.
For a more comprehensive understanding, let's compare data types. In contrast to categorical data, a quantitative variable is numeric and can be measured or counted. It can further be subdivided into discrete or continuous data. Quantitative discrete data involve counts of items or occurrences (e.g., the number of classes you take per school year), while quantitative continuous data involve measurements that can take on any value within a given range (e.g., the weights of soups measured in ounces).
Returning to the student's survey question about fruit and vegetable consumption, it is evident that the data collected does not involve counting or measuring numerical values, but rather involves placing respondents into categories based on their dietary habits. Therefore, the variable in question is indeed categorical.
WILL GIVE BRAINLIEST PLS ANSWER
/Given: ABCD is a rhombus, m∠A = 70°
Find: (AREA OF CIRCLE) / (AREA OF RHOMBUS)
Answer:
Step-by-step explanation:
Check the attachment the solution of the work is given there
Answer: 0.74
Step-by-step explanation:
Let h = rhombus' height
Looking at the attachment, we see that the circle has an area of [tex]\pi *(\frac{h}{2}) ^{2}[/tex]
The rhombus has an area of [tex]\frac{h^2}{sin(70°)}[/tex]
because the base is [tex]\frac{b}{sin(90)} = \frac{h}{sin(70)}[/tex]
due to the law of sines
Thus, Area Circle / Area Rhombus is
[tex]\frac{(\pi(\frac{h}{2})^2)}{(\frac{h^2}{sin(70)}) } = 0.74[/tex]
Roxanne is planning to enclose her right triangular shaped garden with a fence. How many
feet of fencing does she need to enclose her entire garden if the length of her garden
measures 19 feet and the hypotenuse of her garden measures 33 feet? Round your answer to
the nearest tenth of a foot.
**Remember... to find the perimeter of an object, you must ADD the lengths of all sides.
Answer:
The perimeter of Roxanne's right triangular garden is 79 feet.
Step-by-step explanation:
Given,
Length of 1 side = 19 feet
Hypotenuse = 33 feet
We have to find out the perimeter of the triangular garden.
Solution,
Since the garden is in shape of right triangle.
So we apply the Pythagoras theorem to find the third side.
"In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides".
So framing in equation form, we get;
[tex]33^2=19^2+(third\ side)^2\\\\1089=361+(third\ side)^2\\\\(third\ side)^2=1089-361\\\\(third\ side)^2=728[/tex]
Now taking square root on both side, we get;
[tex]\sqrt{(third\ side)^2} =\sqrt{728} \\\\third\ side=26.98\approx27\ ft[/tex]
Now we know that the perimeter is equal to sum of all the three side of a triangle.
Perimeter = [tex]19+27+33=79\ ft[/tex]
Hence The perimeter of Roxanne's right triangular garden is 79 feet.
3x to the power of two minus x
Factor by gcf
Answer:
After factorizing the given expression we get the value as [tex]x(3x-1)[/tex].
Step-by-step explanation:
Given:
[tex]3x^2-x[/tex]
We need to factorize the given expression using GCF.
Solution:
[tex]3x^2-x[/tex]
Now GCF means Greatest common factor.
From the given 2 numbers we need to find the greatest common factor.
[tex]3\times x\times x- 1 \times x[/tex]
In the given expression GCF is 'x'.
Hence we can say that;
[tex]x(3x-1)[/tex]
Hence After factorizing the given expression we get the value as [tex]x(3x-1)[/tex].
*50 POINTS -- FRESHMEN ~ ALGEBRA I *
Large boxes weigh 75 pounds, and small boxes weigh 40 pounds.
a. Write an inequality that represents the numbers of large, x, and small, y, boxes a 200-pound delivery person can take on the elevator.
b. Select the reason(s) why some solutions of the inequality might not be practical in real life.
>The number of boxes must be a whole number.
>The number of boxes must be a rational number.
>It is unlikely that one person will carry 20 large boxes.
>It is unlikely that one person will carry 45 small boxes.
For a, I got 75x + 40y ≤ 200 --- I got it wrong but I'm not sure why?
The maximum weight of boxes that can be placed into the elevator is:
[tex]\to 2000 - 200 = 1800 \ lbs[/tex]
(the load limit is the weight of a delivery person). Small crates weigh 40 pounds, whereas large boxes weigh 75 pounds.As a result, [tex]40X + 75Y = 1800[/tex].It should be noted that Y must be an even integer for the equivalence to hold, whereas X might be odd or even because 40X is always even.
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The solution set of a linear system whose augmented matrix is [a b c d] is the same as the solution set of Ax = d, where A = [a b c]. Note: a, b, c, d are all column vectors.True/false
Answer:
True
Step-by-step explanation:
First statement
[a b c | d][x]
[a b c]x=d
ax+bx+cx=d
Second statement
Ax=d
Given that A = [a b c]
[a b c]x=d
ax+bx+cx=d
ax+bx+cx=d
Then, they are going to have the same solutions
The statement is false. The solution sets for the augmented matrix [a, b, c, d] and the matrix equation Ax = d (where A = [a, b, c]) are not the same unless 'd' is consistently a column vector with 'a', 'b', 'c'.
Explanation:The statement presented in the question is false. When we talk about a linear system, an augmented matrix generally pairs a coefficient matrix with an answer matrix. This would look like [A|d], where 'A' would be a matrix, and 'd' is the constants column vector.
Conversely, Ax = d is a matrix equation where 'A' is again the coefficient matrix, 'x' is the variable matrix, and 'd' is the constants column vector.
In your provided augmented matrix, [a b c d], unless 'd' is a consistent column vector with the other column vectors, it can't be virtually the same as the matrix system Ax = d where A = [a b c] because the augmented matrix [a b c d] would mean that A = [a b c] and d = [d].
Unless 'd' is mathematically consistent with the column vectors 'a', 'b', and 'c', the solution sets would not be the same.
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Define a function roll_hundred_pair() that produces a histogram of the results of 100 rolls of two 6-sided dice
Answer:
The code is attached. I used python to define the function and matplotlib library to plot the histogram.
Step-by-step explanation:
I defined a function called roll_hundred_pairI imported matplotlib.pyplot as plt and random I defined a list called diceI created an empty list to collect dice resultsI simulated 100 dice roll using a loop and random.sample finally I plot the histogram using plt.hist methodA rectangle has a perimeter of 50 m and a side length of L.
a. Express the other dimension of the rectangle in terms of L.
Answer: The other dimension can be expressed as
(50 - 2L)/2
Step-by-step explanation: First and foremost, we would let the other dimension be represented by B. Then, the perimeter of a rectangle is measured as L+L+B+B or better put;
Perimeter = 2L + 2B
Where L is the measurement of the longer side and B is the measurement of the shorter side.
In this case the perimeter of the rectangle measures 50m, and this can now be written as
50 = 2L + 2B
Subtract 2L from both sides of the equation
50 - 2L = 2L - 2L + 2B
50 -2L = 2B
Divide both sides of the equation by 2
(50 - 2L)/2 = B
Answer:
25-L
Step-by-step explanation:
Let W represent the other side length. The perimeter (P) of the rectangle is ...
P = 2(W+L)
Solving for W, we get ...
P/2 = W+L
P/2 -L = W
Filling in the given value for P, we find ...
W = 50/2 -L = 25 -L
The other dimension is (25-L) meters.
Drag each expression to the box that describes the expression.
The drag force can be mathematically expressed as Fd = 0.5 × ρ × v^2 × A × Cd, where Fd is the drag force, ρ is the density of the fluid, v is the velocity of the object, A is the reference area, and Cd is the drag coefficient.
Explanation:The drag force can be mathematically expressed as:
Fd = 0.5 × ρ × v2 × A × Cd
Where:
Fd is the drag forceρ is the density of the fluidv is the velocity of the objectA is the reference areaCd is the drag coefficientLearn more about Drag force here:https://brainly.com/question/14748915
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Solve for q. [tex]3\left(q+\dfrac 43\right) = 23[/tex]
Final answer:
To solve for q in the equation 3(q + 4/3) = 23, we distribute the 3, subtract 4 from both sides, and then divide by 3 to find that q is approximately 6.33.
Explanation:
To solve for q in the equation 3(q + \dfrac{4}{3}) = 23, we need to apply some basic algebra principles. First, we distribute the 3 into the parentheses.
3q + 3 \times \dfrac{4}{3} = 23
3q + 4 = 23
Now, subtract 4 from both sides to get 3q alone on one side.
3q = 23 - 4
3q = 19
Last, divide both sides by 3 to solve for q.
q = \dfrac{19}{3}
q = 6.333...
Thus, q is approximately equal to 6.33 when rounded to two decimal places.
The value of q is q= 19 / 3.
Let's solve for q in the equation:
3(q+ 3 / 4)=23
We can solve the equation by distributing the terms, adding/subtracting to both sides, dividing both sides by the same factor, and simplifying.
Steps to solve:
1. Distribute the terms:
3q+4=23
2. Add/subtract to both sides:
3q+4−4=23−4
3q=19
3. Divide both sides by the same factor:
3q / 3 = 19 / 3
4. Simplify:
q= 19 / 3
Therefore, the value of q is q= 19 / 3.
Which equation represents a direct variation?
y = 2х
y=x+4
y= 1/2x
y=3/x
Answer:option 1 and option 2 represents a direct variation.
Step-by-step explanation:
A direct variation is one in which, as one variable increases in value, the other variable increases in value. Also as one variable decreases in value, the other variable decreases in value.
Looking at the options,
1) y = 2х
When x = 2, y = 2×2 = 4
When x = 3, y = 2×3 = 6
Therefore, y = 2x is a direct variation.
2) y=x+4
When x = 2, y = 2 + 4 = 6
When x = 3, y = 3 + 4 = 7
Therefore, y = x + 4 is a direct variation.
3) y= 1/2x
When x = 2, y = 1/2×2 = 1/4 = 0.25
When x = 3, y = 1/2×3 = 1/6 = 0.17
Therefore, y = 1/2x is not a direct variation.
4) y = 3/x
When x = 2, y = 3/2 = 1.5
When x = 3, y = 3/3 = 1
Therefore, y = 3/x is not a direct variation.
Zoey wants to cover her bedroom floor with carpet squares.Each square has an area of 1 square foot.Her bedroom measures 12 feet by 14 feet.How many carpet squares does Zoey need?
Answer: Zoey needs 168 square feet if carpet squares.
Step-by-step explanation:
Zoey wants to cover her bedroom floor with carpet squares. Each square has an area of 1 square foot.
The formula for determining the area of a rectangle is expressed as
Area = length × width
Her bedroom measures 12 feet by 14 feet. Therefore, the area of her bedroom would be
12 × 14 = 168 square feet.
Therefore, the number of carpet squares that Zoey needs would be
168/1 = 168 square feet
Answer:
168 squares
Step-by-step explanation:
Each square is 1 foot on a side, so along the 14-foot wall, Zoey will need 14 squares. Altogether, Zoey will need 12 rows of 14 squares, so 12×14 = 168 squares.
Right △ABC has its right angle at C, BC=4 , and AC=8 .
What is the value of the trigonometric ratio?
Drag a value to each box to match the trigonometric ratio.
Answer:
Therefore,
[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]
[tex]\cot B =\dfrac{1}{2}[/tex]
[tex]\csc B = \dfrac{\sqrt{5}}{2}[/tex]
Step-by-step explanation:
Given:
Right △ABC has its right angle at C,
BC=4 , and AC=8 .
To Find:
Cos A = ?
Cot B = ?
Csc B = ?
Solution:
Right △ABC has its right angle at C, Then by Pythagoras theorem we have
[tex](\textrm{Hypotenuse})^{2} = (\textrm{Shorter leg})^{2}+(\textrm{Longer leg})^{2}[/tex]
Substituting the values we get
[tex](AB)^{2}=4^{2}+8^{2}=80\\AB=\sqrt{80}\\AB=4\sqrt{5}[/tex]
Now by Cosine identity
[tex]\cos A = \dfrac{\textrm{side adjacent to angle A}}{Hypotenuse}\\[/tex]
Substituting the values we get
[tex]\cos A = \dfrac{AC}{AB}=\dfrac{8}{4\sqrt{5}}=\dfrac{2}{\sqrt{5}}\\\\Ratinalizing\\\cos A=\dfrac{2\sqrt{5}}{5}[/tex]
[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]
Now by Cot identity
[tex]\cot B = \dfrac{\textrm{side adjacent to angle B}}{\textrm{side opposite to angle B}}[/tex]
Substituting the values we get
[tex]\cot B = \dfrac{BC}{AC}=\dfrac{4}{8}=\dfrac{1}{2}[/tex]
Now by Cosec identity
[tex]\csc B = \dfrac{Hypotenuse}{\textrm{side opposite to angle B}}\\[/tex]
Substituting the values we get
[tex]\csc B = \dfrac{AB}{AC}=\dfrac{4\sqrt{5}}{8}=\dfrac{\sqrt{5}}{2}[/tex]
Therefore,
[tex]cos A=\dfrac{2\sqrt{5}}{5}[/tex]
[tex]\cot B =\dfrac{1}{2}[/tex]
[tex]\csc B = \dfrac{\sqrt{5}}{2}[/tex]
The distribution of the number of people in line at a grocery store has a mean of 3 and a variance of 9. A sample of the numbers of people in line in 50 stores is taken.
(a) Calculate the probability that the sample mean is more than 4? Round values to four decimal places.
(b) Calculate the probability the sample mean is less than 2.5. Round answers to four decimal places.
(c) Calculate the probability that the the sample mean differs from the population mean by less than 0.5. Round answers to four decimal places.
Answer:
a) [tex]P(\bar X >4)=P(Z>\frac{4-3}{\frac{3}{\sqrt{50}}}=2.357)[/tex]
[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]
b) [tex]P(\bar X <2.5)=P(Z>\frac{2.5-3}{\frac{3}{\sqrt{50}}}=-1.179)[/tex]
[tex]P(Z<-1.179)=0.1192[/tex]
c)
[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]
[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the number of people of a population, and for this case we know that:
Where [tex]\mu=3[/tex] and [tex]\sigma=\sqrt{9}=3[/tex]
The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".
From the central limit theorem we know that the distribution for the sample mean [tex]\bar X[/tex] is given by:
[tex]\bar X \sim N(\mu, \frac{\sigma}{\sqrt{n}})[/tex]
And we want to find this probability:
[tex] P(\bar X >4)= P(z> \frac{4-3}{\frac{3}{\sqrt{50}}})[/tex]
And using a calculator, excel or the normal standard table we have that:
[tex]P(Z>2.357)=1-P(Z<2.357) =1-0.9908=0.0092[/tex]
Part b
[tex] P(\bar X <2.5)= P(z> \frac{2.5-3}{\frac{3}{\sqrt{50}}})[/tex]
And using a calculator, excel or the normal standard table we have that:
[tex]P(Z<-1.179)=0.1192[/tex]
Part c
For this case we want this probability:
[tex] P(2.5 < \bar X< 3.5) = P(\frac{2.5-3}{\frac{3}{\sqrt{50}}} <Z<\frac{3.5-3}{\frac{3}{\sqrt{50}}})[/tex]
And using a calculator, excel or the normal standard table we have that:
[tex]P(-1.179<Z<1.179)=P(Z<1.179)-P(Z<-1.179)=0.8808-0.1192=0.7616 [/tex]
The average amount of a nutrient that is known to meet the needs of 50 percent of the individuals in a similar age and gender group is known as the?
Answer:
Estimated Average Requirement (EAR)
Step-by-step explanation:
The Estimated Average Requirement (EAR) is the average amount of daily intake value which is estimated to meet the needs of 50% of the healthy individuals.
The EAR is estimated on the basis of specific conditions of adequacy, and are derived from a careful study of the literature.
The major parameters which is selected for the criterion are reduction of disease risk.
Alton says that he can draw two triangles that are NOT congruent with two pairs of congruent corresponding angles and a congruent included side because he can extend the rays to meet somewhere other than point Q. Is he correct?
Answer:
No because if the Rays meet at a point other than Q the angles will change
Step-by-step explanation:
Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ∈ R if and only if a + d = b + c. Show that R is an equivalence relation.
Answer:
Therefore, we conclude that R is an equivalence relation.
Step-by-step explanation:
We know that a relation on a set is called an equivalence relation if it is reflexive, symmetric, and transitive.
R is refleksive because we have that a+b = a+b.
R is symmetric because we have that a+d =b+c equivalent with b+c =a+d.
R is transitive because we have that:
((a, b), (c, d)) ∈ R ; ((c, d), (e, f)) ∈ R
a+d =b+c ⇒ a-b=c-d
c+f =d+e ⇒ c-d =e-f
we get
a-b=e-f ⇒ a+f=b+e ⇒((a, b), (e, f)) ∈ R.
Therefore, we conclude that R is an equivalence relation.
Roger is having a picnic for 78guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which inequality could be used to determine the number of packages of hot dogs roger must buy?
Question is Incomplete; Complete question is given below;
Roger is having a picnic for 78 guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which inequality could be used to determine how many packages of hot dogs Roger will need to buy?
1) [tex]p \geq 78[/tex]
2) [tex]8p \geq 78[/tex]
3) [tex]8 +p \geq 78[/tex]
4) [tex]78 + p \geq 8[/tex]
Answer:
2) [tex]8p \geq 78[/tex]
Step-by-step explanation:
Given:
Number of guest in the picnic = 78 guest
Number of hot dog each guest will have = 1
Number of hot dogs in each package = 8 hot dogs.
We need to write the In equality used to determine the number of packages of hot dogs roger must buy
Solution:
Let the number of packages be 'p'.
First we will find the total number of hot dogs required.
so we can say that;
total number of hot dogs required is equal Number of guest in the picnic multiplied by Number of hot dog each guest will have.
framing in equation form we get;
total number of hot dogs required = [tex]78\times 1 =78[/tex]
Now we can say that;
Number of hot dogs in each package multiplied by number of packages should be greater than or equal to total number of hot dogs required.
framing in equation form we get;
[tex]8p\geq 78[/tex]
Hence The In equality used to determine the number of packages of hot dogs roger must buy is [tex]8p\geq 78[/tex].
Write a piece wise function that models this function
The answer is
[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
========================================================
Here's how I got that answer:
Start with the piecewise definition for y = |x|.
[tex]g(x) = \begin{cases}x \text{ if }x \ge 0 \\ -x \text{ if }x < 0\end{cases}[/tex]
Everywhere you see an 'x', replace it with x+2
[tex]g(x+2) = \begin{cases}x+2 \text{ if }x+2 \ge 0 \\ -(x+2) \text{ if }x+2 < 0\end{cases}[/tex]
[tex]g(x+2) = \begin{cases}x+2 \text{ if }x \ge -2 \\ -x-2 \text{ if }x < -2\end{cases}[/tex]
Now tack on "-4" at the end of each piece so that we shift the function down 4 units
[tex]g(x+2)-4 = \begin{cases}x+2-4 \text{ if }x \ge -2 \\ -x-2-4 \text{ if }x < -2\end{cases}[/tex]
[tex]g(x+2)-4 = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
[tex]f(x) = \begin{cases}x-2 \text{ if }x \ge -2 \\ -x-6 \text{ if }x < -2\end{cases}[/tex]
Check out the attached images below. In figure 1, I graph y = x-2 and y = -x-6 as separate equations on the same xy coordinate system. Then in figure 2, I combine them to form the familiar V shape you see with any absolute value graph.
Given the following functions find the following:
a. Domain
b. The Vertical Asymptote(s)
c. The Horizontal Asymptote
[tex]f(x) = \frac{4x}{2x^{2} +1}[/tex]
The asymptotes are found using the rational function ax^n/ bx^m where n is the degree of the numerator and m is the degree of the denominator.
In the given equation the numerator isn’t raised to any power so n is considered equal to 1. The Demi actor has x raised to the 2nd power so m equals 2.
If n < m then the c axis, y= 0 is the horizontal asymptote.
Also because n is less than m there are no vertical asymptote.
The domain is any real number so the domain would be (-infinity, infinity)
A ball is thrown into the air with an initial upward velocity of 48dt/s. Its height h in feet after t seconds is given by the equation h(t)=-16t^2+48t+4
A: What height will the ball be after 2 seconds?
B: After how many seconds will the ball reach its maximum haight?
C: What is the balls maximum height?
Answer:
Step-by-step explanation:
The equation used to represent the height of the ball, h in feet after t seconds is expressed as
h = -16t^2 + 48t + 4
A) The height of the ball after 2 seconds would be
h = - 16 × 2² + 48 × 2 + 4
h = - 64 + 96 + 4
h = 36 feet
B)The equation is a quadratic equation. The plot of this equation on a graph would give a parabola whose vertex would be equal to the maximum height travelled by the rocket.
The vertex of the parabola is calculated as follows,
Vertex = -b/2a
From the equation,
a = -16
b = 48
Vertex = - - 48/32= 1.5
So the ball will attain maximum height at 1.5 seconds.
C) The maximum height of the ball would be
h = -16 × 1.5² + 48 × 1.5 + 4
h = - 36 + 72 + 4
h = 40 feet
Final answer:
The ball will be at a height of 64 feet after 2 seconds. It will reach its maximum height of 58 feet after 1.5 seconds.
Explanation:
To solve the problem, we need to use the given quadratic equation for the ball's height h(t) = -16t2 + 48t + 4. This equation models the motion of the ball thrown into the air with an initial upward velocity.
A: Height After 2 Seconds
To find the height of the ball after 2 seconds, we substitute t = 2 into the equation:
h(2) = -16(2)2 + 48(2) + 4
h(2) = -16(4) + 96 + 4 = 64 feet
B: Time to Reach Maximum Height
To determine when the ball reaches its maximum height, we need to find the vertex of the parabola, which occurs at t = -b/(2a) where a=-16 and b=48. So t = -48/(2(-16)) = 1.5 seconds.
C: Ball's Maximum Height
To find the ball's maximum height, we substitute t = 1.5 into the height equation:
h(1.5) = -16(1.5)2 + 48(1.5) + 4
h(1.5) = -16(2.25) + 72 + 4 = 58 feet
A shop owner bought some shovels for $5,500. The shovels were sold for $7,300, with a profit of $50 per a shovel. How many shovels were involved?
A. 18.
B. 36.
C. 55.
D. 73.
E. 90.
F. None of these.
Answer:
B.
Step-by-step explanation:
Find the total profit.
P=7300-5500
P=1800
Since each shovel makes up 50 of the profit.
50N=1800
N=1800%2F50
N=36
36 shovels were sold.
Recent research suggests that depression significantly increases the risk of developing dementia later inlife (BBC News, July 6, 2010). In a study involving 949 elderly persons, it was reported that 22% of thosewho had depression went on to develop dementia, compared to only 17% of those who did not havedepression.a. Choose the relevant population and the sample. (You may select more than one answer.) 1. The sample consists of 949 elderly people.2. The population is all elderly people.3. The population is all younger people.4. The sample consists of 949 younger peopleb. Do the numbers 22% and 17% represent the population parameters or sample statistics?
Answer:
a.
Option 1
Option 2
b. Sample statistics
Step-by-step explanation:
a.
The set that includes the list of all possible individual in the interested area of study is termed as population while the portion or a part of population is termed as sample. The given scenario indicates that population is all elderly people from which 949 elderly people are selected for study and so, 949 elderly people are included in sample. So, option 1 and option 2 are correct for indicated scenario.
b.
The percentages 22% and 17% are calculated from 949 elderly people that are indicated as sample. Hence, the percentages 22% and 17% are the measure of sample and so, they represents sample statistics.
Final answer:
The relevant population is all elderly people, and the sample consists of 949 elderly people. The numbers 22% and 17% represent sample statistics.
Explanation:
a. The relevant population is all elderly people, so options 2 and 3 are correct. The sample consists of 949 elderly people, so option 1 is also correct.
b. The numbers 22% and 17% represent sample statistics. The sample statistics are calculated from the data collected in the sample, which in this case is the proportion of elderly persons with depression who went on to develop dementia.
A street is drawn by dilating segment FG¯ about center A with a scale factor greater than 0 but less than 1. Is this an enlargement or a reduction?
Answer: This is an reduction.
Step-by-step explanation:
A dilation a king of transformation that creates an similar image (about a center of dilation) of the actual figure by changing its size with the use of a scale factor(k).It either shrinks or stretches the image.If |k| is greater than 1 then the image is an enlargement .If |k| is less than 1 then the image is an reduction.If |k| is equals to 1 then there is no change in size.Given : A street is drawn by dilating segment [tex]\overline{FG}[/tex] about center A with a scale factor greater than 0 but less than 1.
Then by using (2.) , we can say that this is an reduction.
What is the average miles per gallon (mpg) for all new cars? Using Consumer Reports, a random sample of 35 new cars gave an average of 21.1 mpg.
(a) Identify the variable.
(b) Is the variable quantitative or qualitative?
(c) What is the implied population?
The variable is the average miles per gallon (mpg), which is a quantitative measure. The implied population is all new cars.
Explanation:(a) The variable in this situation is the average miles per gallon (mpg) for all new cars.
(b) The variable is quantitative, as it deals with a numerical measure, i.e., the number of miles a car can travel per a gallon of fuel.
(c) The implied population would be all new cars in general - though it's specified that this is based on a sample from Consumer Reports, which may not cover every single new car in existence.
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All of the students at North High School took a benchmark test. When the administration analyzed the students' grades, they found that the grades were normally distributed and that [blank] of the students received grades with z-scores between 0.15 and 0.85.
Answer:
24.2% students received grades with z-scores between 0.15 and 0.85
Step-by-step explanation:
We are given the following in the question:
The grades of a benchmark test for North High School were normally distributed.
WE have to find the percentage of students that received grades with z-scores between 0.15 and 0.85.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(score between 0.15 < z < 0.85)
[tex]P(0.15 \leq z \leq 0.85)\\\\= P(z \leq 0.85) - P(z \leq 0.15)\\\\\text{Calculating the value from standard normal z-table}\\\\= 0.802 - 0.560 = 0.242 = 24.2\%[/tex]
24.2% students received grades with z-scores between 0.15 and 0.85
In triangle ABC, A=25, c=55 and AB=60. What are the approximate measures of the remaining side lengths of the triangle?
Answer:
[tex]a\approx 31[/tex]
[tex]b\approx 72[/tex]
Step-by-step explanation:
Please find the attachment.
We have been given that in triangle ABC, A=25, C=55 and AB=60. We are asked to find the approximate measures of the remaining side lengths of the triangle.
We will use Law of Sines to solve for side lengths of given triangle.
[tex]\frac{\text{sin}(A)}{a}=\frac{\text{sin}(B)}{b}=\frac{\text{sin}(C)}{c}[/tex], where a, b and c are opposite sides corresponding to angles A, b and C respectively.
Upon substituting our given values, we will get:
[tex]\frac{\text{sin}(25)}{a}=\frac{\text{sin}(55)}{60}[/tex]
[tex]a=\frac{60\text{sin}(25)}{\text{sin}(55)}[/tex]
[tex]a=\frac{60*0.422618261741}{0.819152044289}[/tex]
[tex]a=\frac{25.35709570446}{0.819152044289}[/tex]
[tex]a=30.9552980807967304[/tex]
[tex]a\approx 31[/tex]
Therefore, the measure of side 'a' is approximately 31 units.
We can find measure of angle B using angle sum property as:
[tex]m\angle A+m\angle B+m\angle C=180[/tex]
[tex]25+m\angle B+55=180[/tex]
[tex]m\angle B+80=180[/tex]
[tex]m\angle B=100[/tex]
[tex]\frac{\text{sin}(100)}{b}=\frac{\text{sin}(55)}{60}[/tex]
[tex]b=\frac{60\text{sin}(100)}{\text{sin}(55)}[/tex]
[tex]b=\frac{60*0.984807753012}{0.819152044289}[/tex]
[tex]b=\frac{59.08846518072}{0.819152044289}[/tex]
[tex]b=72.1336967815383509[/tex]
[tex]b\approx 72[/tex]
Therefore, the measure of side 'b' is approximately 72 units.
If there is no relationship (linear or otherwise) between two quantitative variables as observed on a scatterplot, the value of the correlation coefficient, r, is likely to be which of the following?1. Closer to 12. Closer to −13. Closer to 04. Either closer to −1 or 1
Answer:
Option 3) Closer to 0
Step-by-step explanation:
Correlation:
Correlation is a technique that help us to find or define a relationship between two variables. A positive correlation means that an increase in one quantity leads to an increase in another quantity A negative correlation means with increase in one quantity the other quantity decreases. Range of CorrelationValues between 0 and 0.3 tells about a weak positive linear relationship, values between 0.3 and 0.7 shows a moderate positive correlation and a correlation of 0.7 and 1.0 states a strong positive linear relationship.
Values between 0 and -0.3 tells about a weak negative linear relationship, values between -0.3 and -0.7 shows a moderate negative correlation and a correlation value of of -0.7 and -1.0 states a strong negative linear relationship.
A value of 0 tells that there is no correlation between the two variables.Thus, for the given situation, if there is no relationship between two quantitative variables then the value of the correlation coefficient, r, is close to 0