The exact value of cos(θ + π/6) when sin θ = 1/4 and θ is in quadrant II is (-√(45) - 1)/8, found by using the Pythagorean identity and the angle sum identity.
To find the exact value of cos(θ + π/6) when sin θ = 1/4 and θ is in quadrant II, we can use trigonometric identities and properties of angles. First, we use the Pythagorean identity to find cos θ. Since θ is in the second quadrant, cos θ is negative and solving for cos θ gives cos θ = -√(1 - sin² θ) = -√(1 - (1/4)²) = -√(15)/4.
Next, we apply the angle sum identity for cosine, which is cos(α + β) = cos α cos β - sin α sin β. Using this identity, we find:
cos(θ + π/6) = cos θ cos(π/6) - sin θ sin(π/6) = (-√(15)/4)(√(3)/2) - (1/4)(1/2) = -√(45)/8 - 1/8 = (-√(45) - 1) / 8.
The diagonals of a square are _____.
A. perpendicular
B. never congruent
C. parallel
D. sometimes equal
The diagonals of a square are perpendicular to each other. Thus, the correct option is A.
What is a square?It is a polygon with four sides. The total interior angle is 360 degrees. A square's opposite sides are parallel and equal, and each angle is 90 degrees. Its diagonals are all the same length and intersect in the center.
If the angle between the diagonals of the quadrilateral is not equal to the right angle, then the quadrilateral is a rectangle, parallelogram, or trapezoid.
If the angle between the diagonals of the quadrilateral is equal to the right angle, then the quadrilateral is a rhombus and square.
The diagonal of the square intersects at the right angle. Then the diagonals of a square are perpendicular to each other.
Thus, the correct option is A.
More about the square link is given below.
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Write the following statement in if - then form.
Two opposite rays form a straight line.
Which of the following is the hypothesis?
1. if two rays are opposite
2. if a straight line is formed
3. then rays are opposite
The first option is correct:
1.) if to rays are opposite.
what is 13 over 20 in simplest form
please help with math
1. Provide a counterexample for the statement in parentheses. (1 point)“If a figure has four sides, then it is a square”
2. Identify the hypothesis of the statement in parentheses. (1 point)“If two angles form a linear pair, then they are adjacent and supplementary.”
3. Identify the conclusion of the statement in parentheses. (1 point)“If two angles form a linear pair, then they are adjacent and supplementary.
4. Write the statement in parentheses as a biconditional. (1 point)“If two angles form a linear pair, then they are adjacent and supplementary.”
5. Is the statement in parentheses a good definition? Explain. (2 points)“Obtuse angles are fairly large.”
A retailer has determined that the cost C of
ordering and storing x units of a product is
c=6x+900000/x
(a) Write the expression for cost as a single fraction.
(b) Determine the cost for ordering and storing - x=240
units of this product. ...?
In part (a), the expression for cost as a single fraction is (6x^2 + 900000x) / x. In part (b), we substitute x = 240 and simplify the expression to find the cost for ordering and storing 240 units of the product.
Explanation:(a) To write the expression for cost as a single fraction, we need to combine the terms in the numerator. The numerator is 6x + 900000 and the denominator is x. To combine the terms, we multiply 6x by x to get 6x^2, and then add 6x^2 + 900000x in the numerator. Therefore, the expression for cost is (6x^2 + 900000x) / x.
(b) To determine the cost for ordering and storing x = 240 units of this product, we substitute x = 240 into the expression for cost. Plugging in x = 240, we have (6(240)^2 + 900000(240)) / 240. Simplifying this expression gives us the cost for ordering and storing 240 units of the product.
The cost function C = 6x + 900000/x can be written as a single fraction as C = (6x² + 900000) / x. When x = 240, the cost for ordering and storing the units is calculated to be $5190.
The cost C of ordering and storing x units of a product is given by the formula C = 6x + 900000/x. To write this expression as a single fraction, we can find a common denominator and combine the two terms:
C = (6x²+ 900000) / x
For part (b), to determine the cost for ordering and storing x = 240 units of this product, we substitute 240 for x in the expression:
C = (6(240)² + 900000) / 240
Which simplifies to:
C = (6(57600) + 900000) / 240
C = (345600 + 900000) / 240
C = 1245600 / 240
C = 5190
Therefore, the cost for ordering and storing 240 units of this product is $5190.
the three sides of a triangle have lengths 3x, 2x+4, and 5x-8. Find the value of x that makes the triangle equilateral.
Answer:
x=4
Step-by-step explanation:
The value of x that makes the triangle equilateral is 4
How to determine the valueFor an equilateral triangle, all three sides must be equal in length. In this case, the sides are represented by 3x, 2x + 4, and 5x - 8.
To find the value of x that makes the triangle equilateral, set up an equation:
3x = 2x + 4 = 5x - 8
Now, solve for x:
3x = 2x + 4.
Subtract 2x from both sides:
x = 4
Now, check if x = 4 satisfies the other two parts: 2x + 4 and 5x - 8.
For 2x + 4:
2(4) + 4 = 8 + 4 = 12
For 5x - 8:
5(4) - 8 = 20 - 8 = 12
Since all three expressions are equal to 12, the value of x = 4 makes the triangle equilateral.
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What is 20 time 4 equal?
Convert the equation to polar form.
2x=2y ...?
To convert 2x = 2y into polar form, replace x with r cos(theta) and y with r sin(theta), then simplify, resulting in the polar equation cos(theta) = sin(theta) or theta = pi/4 + kpi, where k is an integer.
To convert the equation 2x = 2y to polar form, we need to use the relationship between Cartesian coordinates (x, y) and polar coordinates (r, (theta)). The transformations are given by x = r cos(theta) and y = r sin(theta).
Starting with our original equation 2x = 2y, we can substitute the transformations into the equation, which gives us 2(r cos(theta)) = 2(r sin(theta)). We can divide both sides by 2 to simplify, which results in r cos(theta) = r sin(theta). Since r cannot be 0 (because then both x and y would be 0, which does not satisfy the original equation), we can divide both sides by r, leading to cos(theta) = sin(theta). This equation implies that tan(theta) = 1. When tan(theta) equals 1, theta can be written as theta = arctan(1) or theta = pi/4 + kpi, where k is an integer representing the multiple full rotations around the circle.
the Museum of Science in Boston displays a running total of the U.S. population. on may 11, 1993, the total was increasing at the rate of 1 person every 14 sec. the displayed population figure at 3:45 PM. that day was 257,313,431. assume that the relative growth rate is constant.
A) what is the relative growth rate per year (365 days)?
B) at this rate, what will the U.S population be at 3:45 Pm Boston tiem in May 11, 2001
The relative growth rate per year is approximately 22521.43 people per year. The U.S. population at 3:45 PM Boston time on May 11, 2001, will be approximately 275,333,002 people.
Explanation:The relative growth rate per year can be found by converting the rate of 1 person every 14 seconds to the rate per year. There are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day, so there are 60 * 60 * 24 = 86400 seconds in a day. There are 365 days in a year, so the rate per year is (1 person / 14 seconds) * (86400 seconds / 1 day) * (365 days / 1 year) = 22521.4286 people per year. Therefore, the relative growth rate per year is approximately 22521.43 people per year.
To determine the U.S. population at 3:45 PM Boston time on May 11, 2001, we need to calculate the number of seconds between May 11, 1993, and May 11, 2001, at 3:45 PM. There are 365 days in a year and 24 hours in a day, so there are 365 * 24 = 8760 hours in a year. Since May 11, 1993, to May 11, 2001, is 8 years, there are 8 * 8760 = 70080 hours between the two dates. There are 60 minutes in an hour and 60 seconds in a minute, so there are 60 * 60 = 3600 seconds in an hour. Therefore, there are 70080 * 3600 = 252288000 seconds between the two dates. The population increase rate is 1 person every 14 seconds, so the population increase during this time period is 252288000 / 14 = 18020571.43 people. Adding this to the initial population of 257,313,431 gives a total population of 257,313,431 + 18020571.43 = 275,333,002.43. Therefore, the U.S. population at 3:45 PM Boston time on May 11, 2001, will be approximately 275,333,002 people.
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I've been waiting 4 hours for help answering this. I need help
The formula for rate is r=dt.
Solve for d.
Enter your answer in the box.
Kelli is 3 4 as tall as her brother Ted. Write an expression to describe Kelli's height. Ted's height can be represented using the variable x.
Answer:
Kelli's height = [tex]\frac{3}{4}x[/tex]
Step-by-step explanation:
Here, x represents the Ted's height .
As per the statement:
Kelli is [tex]\frac{3}{4}[/tex] as tall as her brother Ted.
We have to find an expression to describe Kelli's height.
then;
[tex]\frac{3}{4}[/tex] as tall as her brother Ted means [tex]\frac{3}{4}x[/tex]
⇒Kelli's height = [tex]\frac{3}{4}x[/tex]
Therefore, an expression to describe Kelli's height is, [tex]\frac{3}{4}x[/tex]
A recipe calls for 32 ounces of orange juice. How many cups of juice would you need for the recipe?
What speed must you toss a ball straight up so that it takes 4 s to return to you? Show your work.
Solve the following equations for all solutions of x
2sin^2x+3cos-3=0
2 4/15 divided by 2 2/5
Jennifer broke open her piggy bank and found 83 coins in nickels and dimes. If she had $6.95 in all, how many coins of each has she?
Jennifer has 27 nickels and 56 dimes.
Let's use the variables n and d to represent the number of nickels and dimes Jennifer has, respectively.
We can set up two equations based on the given information:
n + d = 83 (equation 1)0.05n + 0.10d = 6.95 (equation 2)Now, we can solve this system of equations.
We can rewrite equation 1 as n = 83 - d and substitute it into equation 2:
0.05(83 - d) + 0.10d = 6.95
4.15 - 0.05d + 0.10d = 6.95
0.05d = 6.95 - 4.15
0.05d = 2.80
d = 2.80 / 0.05
d = 56
Substituting the value of d back into equation 1, we find n = 83 - 56 = 27.
Therefore, Jennifer has 27 nickels and 56 dimes.
The height of a coconut falling from a tree can be represented by the function h(t)=-16t^2 + 24, where h(t) is the height of the coconut, in feet, and t is time, in seconds.
What is the initial height, in feet, of the coconut?
Answer:
The answer is C "The values of h(t) when t = 4 and 5 should be 0."
Gasoline
Costs $3.37 per gallon.mary's father put 9 gallon of gasoline in the tank of his car.how much will the gasoline cost
what is the solution to this inequality?
x/4 < 12
A. x < 48
B. x < 3
C. x > 3
D. x > 48
The results of a survey show that the percent of adults in a certain town who want to change the name of the town is in the interval . What is the point estimate for the percent who want to change the town’s name? What is the poll’s margin of error? Do you think the town is most likely to change its name? Which statistic influenced your answer the most? Explain.
Please help.. what is the function rule for the perimeter P of a building with a rectangular base if the width w is two times the length L?
A.) P=2L B.) P=6L C.) P=6w
Answer:
B.) P = 6LStep-by-step explanation:
The perimeter is the sum of all sides of the figure. In this case, the perimeter of the rectangle would be: [tex]P=W+L+W+L[/tex]; where [tex]W[/tex] is width, and [tex]L[/tex] is length.
According to the problem, the width is two times the length:
[tex]W=2L[/tex]
Replacing this relation in the perimeter equation, we have:
[tex]P=2W+2L\\P=2(2L)+2L\\P=4L+2L\\P=6L[/tex]
Therefore, the perimeter is six times the length of the rectangle. The correct answer is B.
What is two plus two plus two plus two??????
conjugate of 8+4i ...?
What is the lcm of 9,45,81?
Simplify. Show your work.
5 1/3 + (-3 9/18)
Answer: Simplified form is
[tex]1\frac{5}{6}[/tex]
Step-by-step explanation:
Since we have given that
[tex]5\frac{1}{3}-3\frac{9}{18}[/tex]
Now, we will simplify it with the following steps :
[tex]5\frac{1}{3}-3\frac{9}{18}\\\\=\frac{16}{3}-3\frac{1}{2}\\\\=\frac{16}{3}-\frac{7}{2}\\\\=\frac{32-21}{6}\\\\=\frac{11}{6}\\\\=1\frac{5}{6}[/tex]
Hence, simplified form is
[tex]1\frac{5}{6}[/tex]
What is the length of the altitude of the equilateral triangle below
Method 1
Applying the Pythagorean Theorem
we know that
[tex]10^{2}= 5^{2} +a^{2}[/tex]
Solve for a
[tex]100= 25 +a^{2}[/tex]
[tex]a^{2}=100-25[/tex]
[tex]a^{2}=75[/tex]
[tex]a=\sqrt{75}=5 \sqrt{3}\ units[/tex]
therefore
the answer is
the length of the altitude is [tex]5 \sqrt{3}\ units[/tex]
Method 2
we know that
[tex]sin(60\°)=\frac{\sqrt{3}}{2}[/tex] -------> equation A
and
in this problem
[tex]sin(60\°)=\frac{a}{10}[/tex] --------> equation B
equate equation A and equation B
[tex]\frac{\sqrt{3}}{2}=\frac{a}{10}\\\\a=\frac{10\sqrt{3}}{2}\\\\a=5 \sqrt{3}\ units[/tex]
therefore
the answer is
the length of the altitude is [tex]5 \sqrt{3}\ units[/tex]
The probability that a dessert sold at a certain cafe contains chocolate is 73%. The probability that a dessert containing chocolate also contains nuts is 25%. Find the probability that a dessert chosen at random contains nuts given that it contains chocolate. Round to the nearest tenth of a percent.
The probability that a dessert chosen at random contains nuts given that it contains chocolate is approximately 0.34.
Explanation:To find the probability that a dessert chosen at random contains nuts given that it contains chocolate, we can use the formula for conditional probability:
P(N|C) = P(C and N) / P(C)
We are given that the probability that a dessert contains chocolate is 73% or 0.73, and the probability that a dessert containing chocolate also contains nuts is 25% or 0.25. Therefore, P(N|C) = 0.25 / 0.73 ≈ 0.34, rounded to the nearest tenth of a percent.
The probability that a dessert contains nuts given it contains chocolate is approximately 34.3%.
Probability of a dessert containing chocolate = 73% (0.73)
Probability of a dessert containing nuts given it contains chocolate = 25% (0.25)
Probability that a dessert contains nuts given it contains chocolate
Let C be the event that the dessert contains chocolate and N be the event that it contains nuts.
Therefore, the probability that a dessert chosen at random contains nuts given that it contains chocolate is approximately 34.3%.
if there is a 30% chance of sun tomorrow and a 20% chance of wind and no sun, what is the probability that it is windy, given that it is not sunny? Round your answer to the nearest percent.
A. 44%
B. 29%
C. 22%
D. 57%
Answer:
B. 29%
Step-by-step explanation:
if there is a 30% chance of sun tomorrow and a 20% chance of wind and no sun.
what is the probability that it is windy, given that it is not sunny.
A: it is not sunny
P(A)=0.7 (since, chance of being sunny=30% then chance of not being sunny=70%)
B:it is windy
A∩B:it is windy and not sunny
P(A∩B)=0.2 (chance=20%)
B/A:it is windy given that it is not sunny
Baye's theorem states that:
P(A∩B)=P(B/A)×P(A)
0.2=P(B/A)×0.7
⇒ P(B/A)= 2/7=0.286
≈ 29%
Hence, probability that it is windy, given that it is not sunny is:
B. 29%
3x+6y=18. 3y=-3/2x+9 solve as a substitution problem
simplify completely 3x+18/18 ...?