Find the missing coordinates for the given rule.
Given: S(4,5), R(-5,8), T(-2,3)
RULE: rotate clockwise 90-degrees
Answers
The requried coordinates of points S', R', and T' are (5,-4), (8,5), and (3,2).
To find the coordinates of the image points after rotating 90 degrees clockwise, we can use the following formulas:
x' = y
y' = -x
For point S(4,5), the coordinates of the image point S' after rotating 90 degrees clockwise can be found as follows:
x' = y = 5
y' = -x = -4
Therefore, the image point S' is (5,-4).
For point R(-5,8), the coordinates of the image point R' after rotating 90 degrees clockwise can be found as follows:
x' = y = 8
y' = -x = 5
Therefore, the image point R' is (8,5).
For point T(-2,3), the coordinates of the image point T' after rotating 90 degrees clockwise can be found as follows:
x' = y = 3
y' = -x = 2
Therefore, the image point T' is (3,2).
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Related Questions
(8 marks) Find the root of the equation, f(x) = xe^x – 1 using fixed point iteration and Aitken Acceleration, accurate up to machine epsilon of 1 x 10^-5. Use the iteration formula g(x) = e^-x, and start the iteration using xo = 0.
Answers
To find the root of the equation f(x) = xe^x – 1 using fixed point iteration and Aitken Acceleration, accurate up to machine epsilon of 1 x 10^-5, we will use the iteration formula g(x) = e^-x and start the iteration using xo = 0.
1. Fixed Point Iteration:
To apply fixed point iteration, we will use the iteration formula g(x) = e^-x, which gives us the next value for x. The algorithm for fixed point iteration is:
- Start with an initial guess, xo = 0
- Iterate using xn+1 = g(xn) until |xn+1 - xn| < ε, where ε = 1 x 10^-5
Using this algorithm, we get the following iterations:
x0 = 0
x1 = g(x0) = e^0 = 1
x2 = g(x1) = e^-1 ≈ 0.36788
x3 = g(x2) = e^-0.36788 ≈ 0.69315
x4 = g(x3) = e^-0.69315 ≈ 0.50000
x5 = g(x4) = e^-0.50000 ≈ 0.60653
x6 = g(x5) = e^-0.60653 ≈ 0.54520
x7 = g(x6) = e^-0.54520 ≈ 0.57961
x8 = g(x7) = e^-0.57961 ≈ 0.56012
x9 = g(x8) = e^-0.56012 ≈ 0.57114
x10 = g(x9) = e^-0.57114 ≈ 0.56488
After 10 iterations, we get an approximate solution of x ≈ 0.56488, which is accurate up to machine epsilon of 1 x 10^-5.
2. Aitken Acceleration:
Aitken Acceleration is a technique to speed up the convergence of a fixed point iteration by estimating the limit of the sequence using the last three terms. The algorithm for Aitken Acceleration is:
- Start with an initial guess, xo = 0
- Iterate using xn+1 = g(xn) until |xn+1 - xn| < ε, where ε = 1 x 10^-5
- Apply Aitken Acceleration to the sequence {xn} using the formula:
y_n = x_n - (x_n - x_{n-1})^2 / (x_n - 2x_{n-1} + x_{n-2})
- Iterate using y_n until |y_n+1 - y_n| < ε
Using this algorithm, we get the following iterations:
x0 = 0
x1 = g(x0) = e^0 = 1
x2 = g(x1) = e^-1 ≈ 0.36788
x3 = g(x2) = e^-0.36788 ≈ 0.69315
Then, we apply Aitken Acceleration to the sequence {xn}:
y0 = x0 = 0
y1 = x1 = 1
y2 = x2 - (x2 - x1)^2 / (x2 - 2x1 + x0) ≈ 0.56714
y3 = x3 - (x3 - x2)^2 / (x3 - 2x2 + x1) ≈ 0.56408
After 3 iterations, we get an approximate solution of x ≈ 0.56408, which is accurate up to machine epsilon of 1 x 10^-5. Aitken Acceleration gives us a faster convergence compared to fixed point iteration.
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Claim. The standard deviation of pulse rates of adult males is less than 10 bpm. For a random samplo of 135 adult males, the pulse rates have a standard deviation of 9.2 bpm. Find the value of the tes
Answers
The value of the test statistics is 113.42.
To test the claim that the standard deviation of pulse rates of adult males is less than 10 bpm, we will use a chi-square test. Given a random sample of 135 adult males with a standard deviation of 9.2 bpm, let's find the value of the test statistic.
Step 1: State the null and alternative hypotheses.
H0: σ = 10 bpm (null hypothesis)
H1: σ < 10 bpm (alternative hypothesis)
Step 2: Determine the appropriate test statistic.
In this case, we will use the chi-square test statistic: χ² = (n-1)(s²/σ²), where n is the sample size, s is the sample standard deviation, and σ is the population standard deviation.
Step 3: Calculate the test statistic.
χ² = (135-1)(9.2²/10²) = (134)(84.64/100) = 134 * 0.8464 ≈ 113.42
The value of the test statistic is approximately 113.42. This value can be compared to the critical value from the chi-square distribution table with (n-1) degrees of freedom to determine whether to reject or fail to reject the null hypothesis.
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Trisha owns 25 shares of a common stock in a pharmaceutical company. Last month the price of the stock was $35.48 per share. Today, the price of the stock is $27.36. By how much did the value of the stock decrease?
Enter your answers as a number like 105.
Answers
The value of the stock decreased by $203.
We have,
The initial value of the stock is:
= 25 shares X $35.48/share
= $887
The current value of the stock is:
= 25 shares x $27.36/share
= $684
The value of the stock decreased by:
= $887 - $684
= $203
Thus,
The value of the stock decreased by $203.
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sara mcmahon purchased a new car 3 years ago for $24,500.00. the current estimated value is $17,900.00. annual variable costs this year were $895.60. insurance was $1,350.00, registration was $132.50, and loan interest totaled $1,080.00. she drove 12,540 miles this year. compute the cost per mile in dollars. round to the nearest hundredth.
Answers
The cost per mile to the nearest hundredth: $0.28 per mile.
Sara McMahon purchased a new car 3 years ago for $24,500.00, and the current estimated value is $17,900.00. The annual variable costs this year were $895.60, insurance was $1,350.00, registration was $132.50, and loan interest totaled $1,080.00. She drove 12,540 miles this year.
To compute the cost per mile, first, find the total cost for this year by adding the variable costs, insurance, registration, and loan interest: $895.60 + $1,350.00 + $132.50 + $1,080.00 = $3,458.10.
Next, divide the total cost by the number of miles driven: $3,458.10 ÷ 12,540 miles = $0.2757 per mile.
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The manager of a small convenience store does not want her customers standing in long too long prior to a purchase. In particular, she is willing to hire an employee for another cash register if the average wait time of the customers is more than five minutes. She randomly observes the wait time (in minutes) of customers during the day: 3.5 5.8 7.2 1.9 6.8 8.1 5.4 Assume x-bar = 5.53 and s = 0.67. What is the appropriate conclusion at a 5% significance level? a) A new employee does not need to be hired since: .05 < p-value < .10 b) A new employee needs to be hired since: .025 < p-value < .05 c) A new employee does not need to be hired since: .025 < p-value < .05 d) A new employee needs to be hired since: .01 < p-value < .025
Answers
The appropriate conclusion at a 5% significance level is that a new employee needs to be hired since the p-value is less than 0.05.
To test the hypothesis, we will use a one-sample t-test with a null hypothesis that the true population mean wait time is less than or equal to 5 minutes. The alternative hypothesis is that the true population mean wait time is greater than 5 minutes.
Using the given sample data, we calculate the sample mean (x-bar) as 5.53 and the sample standard deviation (s) as 0.67. The sample size is 7.
We calculate the t-statistic using the formula t = (x-bar - mu)/(s/sqrt(n)), where mu is the hypothesized population mean (5) and n is the sample size.
Substituting the values, we get t = (5.53 - 5)/(0.67/sqrt(7)) = 2.44.
Using a t-distribution table with 6 degrees of freedom (n-1), we find the p-value to be 0.03 for a one-tailed test. Since the p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that a new employee needs to be hired to reduce the average wait time.
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Year-round-Recreation sells recreation vechiles (cross country motorcycles to snowmobiles) and has total costs given by
C(e) 2750+30x+2
and the total revenues for Year-round-Recreation is given by
R(z) = 135x
Find the x-values of the break-even points.
The break-even x-value(s) are (separate by commas - order does not matter)
Answers
The break-even x-value for Year-round-Recreation is approximately 26.19.
To find the break-even points for Year-round-Recreation, we need to set the total costs equal to total revenues and solve for x. In this case, the total costs are given by C(x) = 2750 + 30x + 2, and the total revenues are given by R(x) = 135x.
The break-even point is when C(x) = R(x), so:
2750 + 30x + 2 = 135x
Now, we need to solve for x:
1. Subtract 30x from both sides:
2750 + 2 = 105x
2. Subtract 2 from both sides:
2750 = 105x
3. Divide both sides by 105:
x = 2750 / 105
x ≈ 26.19
The break-even x-value for Year-round-Recreation is approximately 26.19.
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PLEASE ANSWER QUICK!!!!! 45 POINTS
Find the probability of exactly one successes in five trials of a binomial experiment in which the probability of success is 5%
round to the nearest tenth
Answers
Answer:
We can use the formula for the probability mass function of a binomial distribution:
P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)
Where:
n = number of trials
k = number of successes
p = probability of success
In this case, n = 5, k = 1, and p = 0.05. Plugging these values into the formula, we get:
P(X = 1) = (5 choose 1) * 0.05^1 * (1 - 0.05)^(5-1) ≈ 0.23
Rounding to the nearest tenth, the probability of exactly one success in five trials with a 5% probability of success is approximately 0.2 or 20%.
Step-by-step explanation:
To encourage a student in his work on rates and ratios, his teacher promises to pay him 70 cents for every correct problem he solves. However, for every problem where he gives an incorrect answer, the teacher will take 40 cents off him! Amazingly, at the end of 33 problems completed, neither owes anything to the other. So how many problems did the student solve correctly? Investigate this fully (give evidence) and clearly show how you arrive at your solution.
Answers only will be awarded O marks
Answers
The student solved 12 problems correctly.
To find out how many problems the student solved correctly, we can use the given information and set up an equation using the terms "correct problems" and "incorrect problems."
Let x represent the number of correct problems and y represent the number of incorrect problems. We know the following:
1. The total number of problems completed is 33, so x + y = 33.
2. The teacher pays 70 cents for correct problems and takes 40 cents for incorrect problems, and neither owes anything to each other. So, 70x - 40y = 0.
Now, we'll solve this system of equations step-by-step:
Step 1: Solve the first equation for x: x = 33 - y.
Step 2: Substitute the expression for x in the second equation: 70(33 - y) - 40y = 0.
Step 3: Simplify and solve for y: 2310 - 70y - 40y = 0 => 2310 - 110y = 0 => y = 21.
Step 4: Substitute the value of y back into the equation for x: x = 33 - 21 => x = 12.
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The first three terms of a sequence are given. Round to the nearest thousandth (if necessary). find the 7th term 18,6,2
Answers
The 7th term of the sequence is 0.297
We have,
To find the 7th term, we need to know the common ratio.
We can find the common ratio by dividing any term by the previous term.
Common ratio = 6/18 = 1/3
Now we can use the formula for the nth term of a geometric sequence:
[tex]a_n = a_1 \times r^{n-1}[/tex]
where a(1) is the first term and r is the common ratio.
So, for this sequence:
a(1) = 18
r = 1/3
To find the 7th term:
a(7) = 18 x (1/3)^{7 - 1}
= 0.297
Rounding to the nearest thousandth:
Thus,
The 7th term of the sequence is 0.297
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Determine whether the following statement pattern is a tautology or a contradiction or contingency:(p→q)∨(q→p)
Answers
The given statement pattern is a tautology.
To determine whether the following statement pattern is a tautology, a contradiction, or a contingency: (p→q)∨(q→p), follow these steps:
1. Write out the truth table for p and q:
p | q
-----
T | T
T | F
F | T
F | F
2. Calculate the truth values for (p→q) and (q→p) using the implication rule (p→q is false only when p is true and q is false):
(p→q) | (q→p)
-----------
T | T
F | T
T | F
T | T
3. Finally, calculate the truth values for the given statement pattern (p→q)∨(q→p) using the disjunction rule (a disjunction is true if at least one of the statements is true):
(p→q)∨(q→p)
---------
T
T
T
T
Since the statement pattern (p→q)∨(q→p) is true for all possible truth values of p and q, it is a tautology.
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pls help me i need to show work and i need it asap
Answers
(1) The two triangles are similar because they have equal angles.
(2) Triangle QRS is similar to triangle QLM because they have equal angles.
(3) Both triangles are similar and the value of x is 21.
What are the measure of the triangles?Two triangles are said to be similar if they have equal sides, equal angles or both.
The missing angles of the triangles for the question is calculated as;
Bigger triangle; missing angle = 180 - (44 + 46) = 90
Smaller triangle; missing angle = 90 - 46 = 44⁰
Both triangles are similar.
For the second question; triangle QRS is similar to triangle QLM because angle R is equal to angle L, and also they have common angle Q, which implies that angle S must be equal to angle L.
For third question, the triangles are similar because their corresponding angles are equal.
The value of x is calculated as;
48 + 4x + (180 - (56 + 76)) = 180 (sum of angles on a straight line)
48 + 4x + 48 = 180
4x = 84
x = 84/4
x = 21
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Kim has 2,835 comic books. He must pack them into boxes to ship to a comic book store. Each box holds 45 comic books. How many boxes will he need to pack all of the books. ?
Answers
Answer:
The answer to your problem is, 63
Step-by-step explanation:
So we know that he has 2,835 comic books. He is also going to put them in boxes to ship it in a book store.
1 Box = 45 Comic Books
So in order to solve the problem we need to divide:
The expression includes:
2,835 ÷ 45 = 63
Thus the answer to your problem is, 63
a machine is used to fill 1-liter bottles of a type of soft drink. we can assume that the output of the machine approximately follows a normal distribution with a mean of 1.0 liter and a standard deviation of .01 liter. the firm uses means of samples of 25 observations to monitor the output, answer the following questions: determine the upper limit of the control chart such that it will include roughly 97 percent of the sample means when the process is in control. (3 decimal points are required)
Answers
The upper limit of the control chart such that it will include roughly 97 percent of the sample means when the process is in control is 1.008.
For a process with a normal distribution, the mean of the sample means is equal to the population mean and the standard deviation of the sample means is equal to the population standard deviation divided by the square root of the sample size. In this case, the mean of the output is 1.0 liter and the standard deviation is 0.01 liter, so the standard deviation of the sample means is 0.01 / √25 = 0.002.
To construct a control chart for the sample means, we need to determine the upper and lower control limits such that the process is in control when the sample means fall within these limits. Assuming the process is in control, we want to find the upper limit such that roughly 97% of the sample means fall below this limit.
Using the standard normal distribution, the Z-score corresponding to the 97th percentile is approximately 1.88.
Therefore, the upper control limit is 1.0 + 1.88(0.002) = 1.008. Any sample mean above this limit should be investigated for potential process issues.
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Find m∠T in parallelogram QRST.
Answers
The unknown angle of the parallelogram is as follows:
m∠T = 63 degrees
How to find the angle of a parallelogram?A parallelogram is a quadrilateral with opposite sides parallel to each other and opposite congruent to each other.
Therefore, the opposite angles of a parallelogram are equal. Consecutive angles are supplementary angles to each other.
Hence,
10w + 53 + 17w + 100 = 180
27w + 153 = 180
27w = 180 - 153
27w = 27
divide both sides by 27
w = 27 / 27
w = 1
Therefore,
m∠T = 10(1) + 53 = 63 degrees
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Help please and thank youuuuuu
Answers
The value of x in the rectangular prism is 9 inches.
How to find the height of the rectangular prism?The height of the rectangular prism can be found as follows:
The volume of the rectangular prism is 153 inches cube.
Therefore,
volume of the rectangular prism = lwh
where
l = lengthh = heightw = widthTherefore,
volume of the rectangular prism = 8.5 × 2 × x
153 = 17x
divide both sides by 17
x = 153 / 17
x = 9 inches
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Enter your answer and show all the steps that you use to solve this problem in the space provided. Use the 30°-60°-90° Triangle Theorem to find the answer.
Answers
Answer:
x = 5√3, and y = 10.
In a 30°-60°-90° triangle, the length of the longer leg is √3 times the length of the shorter leg, and the length of the hypotenuse is twice the length of the shorter leg.
Find the endpoints of the t distribution wit 2.5% beyond them in each tail if the samples have sizes n1 = 15 and n2 = 22
Answers
The endpoints of the t-distribution with 2.5% beyond them in each tail for the given sample sizes are approximately -2.0301 and 2.0301.
To find the endpoints of the t-distribution with 2.5% beyond them in each tail for the given sample sizes, follow these steps:
1. Determine the degrees of freedom: Since you have two samples with sizes n1 = 15 and n2 = 22, the degrees of freedom (df) will be (n1 - 1) + (n2 - 1) = 14 + 21 = 35.
2. Find the t-value corresponding to the 2.5% tail probability: Using a t-distribution table or an online calculator, look for the t-value that corresponds to a cumulative probability of 0.975 (since you want 2.5% in each tail, and the remaining 95% is between the tails). For df = 35, the t-value is approximately 2.0301.
3. Determine the endpoints: The endpoints of the t-distribution will be the positive and negative t-values found in step 2. So, the endpoints are approximately -2.0301 and 2.0301.
Thus, the endpoints of the t-distribution with 2.5% beyond them in each tail for the given sample sizes are approximately -2.0301 and 2.0301.
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In the diagram below, QP is tangent to a circle with the centre O. Rs is a straight line. T is a point on the circle. PS bisects TPQ and SPQ=22°
Answers
Answer:
In the given diagram, QP is tangent to a circle with centre O, RS is a straight line, T is a point on the circle, and PS bisects TPQ. We know that SPQ = 22°. Let's try to find out the value of the angle TPQ.
Since QP is tangent to the circle, the angle between RS and QP (i.e., angle RQP) is equal to the angle between QP and the radius drawn to the point of tangency (i.e., angle QOT). So, we can say that:
angle RQP = angle QOT
Also, since PS bisects TPQ, we can say that:
angle TPS = angle TPQ / 2
Now, let's consider the triangle TPQ. We know that:
angle TPQ + angle TQP + angle PTQ = 180° [Sum of angles in a triangle]
Substituting the values we have:
angle TPQ + angle TQP + (angle TPS + angle SPQ) = 180°
angle TPQ + angle TQP + (angle TPQ/2 + 22°) = 180°
Multiplying both sides by 2 to eliminate the fraction:
2(angle TPQ) + 2(angle TQP) + angle TPQ + 44° = 360°
Simplifying:
3(angle TPQ) + 2(angle TQP) = 316°
We don't know the values of angle TPQ and angle TQP, so we can't solve this equation exactly. However, we do know that these angles are both less than 180° (since they are angles in a triangle). Therefore, we can try some values for angle TPQ (let's call it x) and see if we can find a corresponding value for angle TQP that satisfies the equation.
If we take x = 40°, then we get:
3(40°) + 2(angle TQP) = 316°
120° + 2(angle TQP) = 316°
2(angle TQP) = 196°
angle TQP = 98°
Now, we can use the fact that angle TPS = angle TPQ / 2 to find angle TPS:
angle TPS = x/2 = 20°
Finally, we can use the fact that PS bisects TPQ to find angle PQT:
angle PQT = angle TPS
Step-by-step explanation:
To find the volume of a rectangular prism, Harris multiplies the area of the base times the height. The area of the base is (x + 4) square inches for some value of x. The height is (2x + 3) inches. What is the volume, in cubic inches, of the rectangular prism? A2x² +12x
B2x² +11x +12
C2^2x + 7x+12
D11x
Answers
Answer:
The correct answer is option B: 2x^2 + 11x + 12 cubic inches
Step-by-step explanation:
The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width (or base), and h is the height of the prism.
Given that the area of the base is (x + 4) square inches and the height is (2x + 3) inches, we can substitute these values into the formula to find the volume:
V = (x + 4)(2x + 3)
Now, we can multiply the binomials using the distributive property:
V = 2x^2 + 3x + 8x + 12
V = 2x^2 + 11x + 12
So, the correct answer is option B: 2x^2 + 11x + 12 cubic inches.
Multiply: (3x−5)(−x+4)
Applying the distributive property, the expression becomes (3x)(−x)+(3x)(4)+(−5)(−x)+(−5)(4).
What is the simplified product in standard form?
x2+
x+
Answers
Answer:
-3x^2 + 12x + 5x - 20 = -3x^2 + 17x - 20
Step-by-step explanation:
(-3x - 5)(-x + 4) is a binomial expression, where (-3x - 5) is one expression and (-x + 4) is the other.
As the text eludes to, we can multiply binomial expressions using the FOIL method, where you multiply the first terms (3x and -x), outer terms (3x and 4), the inner terms (-5 and -x), and the last terms (-5 and 4)
This is how you get
(3x)(-x) + (3x)(4) + (-5)(-x) + (-5)(4)
Now, multiply the terms and combine like terms:
[tex](3x)(-x)+(3x)(4)+(-5)(-x)+(-5)(4)\\-3x^2+12x+5x-20\\-3x^2+17x-20[/tex]
Which is a counterexample for the conditional statement? If two positive numbers are multiplied together, then the product will be greater than both of the two positive numbers. 2 x 4 5 x (−3)
Answers
The counterexample is 2/3 x 9 if two positive numbers are multiplied together and the result is bigger than either of the two positive numbers. d is the right answer, thus.
It is defined as the method through which we multiply, divide, add, and subtract numerical quantities. It contains the basic operators +, -,, and.
Multiplication is a useful tool for carrying out many common tasks, such as computing area, sales tax, and other geometric measurements.
The result will be greater than each of the two positive numbers if a two positive numbers when multiplied together.
If a two positive numbers in the stated condition are x and y,
xy > x
xy> y
The two figures are found to be 2/3 and 9.
=2/3 x 9 =6
Therefore, 2/3 x 9 will serve as the example that refutes the assertion "If two positive numbers when multiplied together, then perhaps the product would be greater than either of the two positive numbers
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The complete question is
The correct question is-
Which is a counterexample for the conditional statement?If two positive numbers are multiplied together, then the product will be greater than both of the two positive numbers.
a. 2 x 4
b. 5x(-3)
c. x
d. 2/3x9
Answer: a
Step-by-step explanation:
The pizza box measures 2/3 feet wide by 4/5 feet long. What is the area of the pizza box
Answers
The area of the pizza box measuring 2/3 feet wide by 4/5 feet long is 8/15 square feet.
The shape of the pizza box is a rectangle. The rectangle is a quadrilateral with opposite sides parallel and equal with an equal angle and of 90°.
The area of a rectangle is considered as:
A = L * B
where L is the length
B is the breadth
Given in the question,
L = 4/5 feet
B = 2/3 feet
The area is calculated by multiplying the fractions. For the multiplication of fractions, we multiply the numerators and denominators separately. And final answer is calculated by simplifying the resulting fraction.
A = 4/5 * 2/3
= 4*2 / 5*3
= 8/15 square feet
Thus, the pizza box has an area of 8/15 square feet
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write the radicand as the product of a perfect cube first and a factor that does not contain a perfect cube (second)
Answers
The number or expression underneath the top line of the symbol is called the radicand. The cube root symbol is a grouping symbol, meaning that all operations in the radicand are grouped as if they were in parentheses.
To write the radicand as the product of a perfect cube (first) and a factor that does not contain a perfect cube (second), follow these steps:
1. Identify the radicand in the given expression. The radicand is the number or expression inside the cube root symbol.
2. Determine the prime factors of the radicand by breaking it down into its smallest prime factors.
3. Group the prime factors into sets of three identical factors. These sets will form the perfect cube factors.
4. Multiply the sets of three factors together to form the perfect cube part of the product.
5. Multiply any remaining factors together to form the factor that does not contain a perfect cube.
6. Write the radicand as the product of the perfect cube (step 4 result) and the factor that does not contain a perfect cube (step 5 result).
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Given the demand function is D(x) = (x - 5)^2 and supply function is S(x) = x^2 + x + 3. Find each of the following: a) The equilibrium point. B) The consumer surplus at the equilibrium point. Explain what the answer means in a complete sentence using the definition of consumer surplus. C) The producer surplus at the equilibrium point. Explain what the answer means in a complete sentence using the definition of producer surplus
Answers
a) The equilibrium point is x = 2 or x = 8
b) The consumer surplus equilibrium point is $6,062.67
c) The producer surplus equilibrium point is $13,208.67
a) To find the equilibrium point, we need to set the demand function equal to the supply function and solve for x:
D(x) = S(x)
[tex](x - 5)^2 = x^2 + x + 3[/tex]
Expanding the left side and simplifying, we get:
[tex]x^2 - 10x + 22 = 0[/tex]
Using the quadratic formula, we get:
[tex]x = (10[/tex] ± [tex]\sqrt{36})/ 2[/tex]
[tex]x = 5[/tex] ± [tex]3[/tex]
[tex]x = 2[/tex] or [tex]x = 8.[/tex]
b) To find the consumer surplus at the equilibrium point, we need to calculate the area under the demand curve and above the equilibrium price, which is given by the supply curve. Since we have two possible equilibrium points, we need to check both of them to see which one gives us a positive consumer surplus.
For [tex]x = 2[/tex], the equilibrium price is given by [tex]S(2) = 11[/tex], which is above the demand curve. Therefore, there is no consumer surplus at this equilibrium point.
For [tex]x = 8[/tex], the equilibrium price is given by [tex]S(8) = 75[/tex], which is below the demand curve. Therefore, the consumer surplus is given by the area under the demand curve and above the price of 75:
[tex]∫[75, 8] (x - 5)^2 dx = [(x - 5)^3 / 3][/tex] from 8 to 75
≈[tex]6,062.67[/tex]
This means that at the equilibrium point x = 8, consumers are willing to pay a total of approximately $6,062.67 more than what they actually pay.
c) To find the producer surplus at the equilibrium point, we need to calculate the area under the equilibrium price and above the su
For x =2 supply curve. Again, since we have two possible equilibrium points, we need to check both of them to see which one gives us a positive producer surplus.
2, the equilibrium price is given by [tex]S(2) = 11,[/tex] which is above the demand curve. Therefore, there is no producer surplus at this equilibrium point.
For x = 8, the equilibrium price is given by[tex]S(8) = 75[/tex], which is below the demand curve. Therefore, the producer surplus is given by the area above the supply curve and below the price of 75:
∫[tex][8, 75] (75 - x^2 - x - 3) dx = [(75x - x^3/ 3 - x^2 / 2 - 3x)][/tex] from 8 to 75
≈ [tex]13,208.67[/tex]
This means that at the equilibrium point x = 8, producers receive a total of approximately $13,208.67 more than their costs.
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Three tennis balls are stored in a cylindrical container with a height of 8.2 inches and a radius of 1.32 inches. The circumference of a tennis ball is 8 inches. Find the amount of space within the cylinder not taken up by the tennis balls. Round your answer to the nearest hundredth.
Answers
The amount of space within the cylinder not taken up by the tennis balls is 18.9 [tex]inches^3[/tex]
The volume of a tennis ball:
The circumference of the tennis ball is 8 inches.
The tennis ball is the form of sphere whose circumference is given by formula [tex]2\pi r[/tex] , where r is the radius.
Thus, if r is the radius then according to condition,
[tex]2\pi r[/tex] = 8 or
r = 8/2[tex]\pi[/tex] inches.
Now, the volume of the sphere of radius r is [tex]\frac{4}{3}\pi r^3[/tex] hence, find the volume of the given tennis ball by substituting r = 8/2[tex]\pi[/tex] inches in [tex]\frac{4}{3}\pi r^3[/tex] and simplify:
Volume = [tex]\frac{4}{3} \pi[/tex] × [tex](\frac{8}{2\pi } )^3[/tex]
Volume = 8.65 [tex]inches^3[/tex]
Hence the required volume of the tennis ball is 8.65 [tex]inches^3[/tex]
The volume of three tennis balls is (3 × 8.65) [tex]inches^3[/tex] = 25.96 [tex]inches^3[/tex]
Find the volume of the cylinder:
The volume of the cylinder with radius r units and height h units is given by [tex]\pi r^2h[/tex] Hence the volume of the given cylinder with radius 1.32 inches , 8.2 height inches is:
[tex]\pi (1.32)^2[/tex] × 8.2
= 3.14 × [tex](1.32)^2[/tex] × 8.2
= 44.86 [tex]inches^3[/tex]
Hence the volume of the cylinder is 44.86 [tex]inches^3[/tex]
Find the amount of space within the cylinder not taken up by the tennis balls.
The required volume can be obtained by subtracting the volume three tennis balls from the volume of the cylinder as follows:
Volume of cylinder - volume of three tennis balls = (44.86 - 25.96) = 18.9 [tex]inches^3[/tex]
Hence, the amount of space within the cylinder not taken up by the tennis balls is 18.9 [tex]inches^3[/tex].
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Determine the equation of any asymptotes in the graph of : (Help!ASAP!)
F(x)= x+3/ x^2-x-12
(Steps by steps)
Answers
The equations of the asymptotes for the graph of F(x) are Vertical asymptote at x = 4 and Horizontal asymptote at y = 0.
To find the equations of the asymptotes, we need to examine the behavior of the function as x gets very large or very small.
First, let's factor the denominator of the function
F(x) = (x + 3) / (x - 4)(x + 3)
Notice that (x + 3) appears in both the numerator and denominator, and therefore can be cancelled out, leaving
F(x) = 1 / (x - 4)
Now, as x gets very large or very small, the value of F(x) approaches 0. However, we can see that as x approaches 4, the denominator of F(x) approaches 0, which means F(x) approaches infinity or negative infinity, depending on which side of x = 4 we approach from.
Therefore, we have a vertical asymptote at x = 4.
To find any horizontal asymptotes, we need to examine the behavior of the function as x approaches infinity or negative infinity. Since the degree of the numerator and denominator are the same (both 1), we can find the horizontal asymptotes by looking at the ratio of the leading coefficients
F(x) = (x + 3) / (x - 4)(x + 3)
As x approaches infinity or negative infinity, the denominator becomes dominated by the highest degree term, x². Therefore
F(x) ≈ (1/x²) / (1 - 4/x + 3/x²)
As x approaches infinity or negative infinity, the terms with x in the denominator become negligible compared to the constant term. Therefore
F(x) ≈ (1/x²) / (1 + 0 + 0) = 1/x²
Thus, we have a horizontal asymptote at y = 0.
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Use the data given in the table below to compute the probability that a randomly chosen voter from the survey will satisfy the following. Round to the nearest hundredth.
The voter is under 50 years old.
Answers
The probability that a randomly chosen voter from the survey is under 50 years old is 0.75
Computing the probability of randomly chosen a voterFrom the question, we have the following parameters that can be used in our computation:
The table of values
Where we have
Voters under 50 years old = 847 + 804 + 773
Total = 3228
So, the required probability is
P = (847 + 804 + 773)/3228
Evaluate
P = 0.75
Hence, the probability is 0.75
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Complete the following description for a graph that shows the distance Sam travels over time when she
runs at a constant rate.
A line that starts at
with a constant (select)
slope.
D
Answers
A line starts at 0 with a constant positive slope.
The graph that shows the distance Sam travels over time is given below.
Here the graph starts at the point (0, 0).
So the line starts at 0.
Now the line is moving in such a way that as time increases, the distance travelled also increases.
So the slope is positive.
Hence the complete description is that line starts at 0 with a constant positive slope.
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Holt Park is divided into two sections. The swing section is 8 yards long and has an area of 112 square yards. The playground section has the same length as the swing section, but it is 3 yards wider. What is the total area of Holt Park?
Answers
A person places $479 in an investment account earning an annual rate of 8. 2%, compounded continuously. Using the formula V = Pe^{rt}V=Pe rt , where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 12 years
Answers
Continuous-compounding is a method of calculating interest where the interest is added to the principal continuously.
instead of being added at regular intervals (such as monthly or annually). This means that the interest is compounded an infinite number of times-over the year, resulting in a higher effective interest rate than other compounding methods.
In this scenario, the person has invested [tex]$479[/tex] in an account that earns an annual interest rate of [tex]8.2%[/tex] compounded continuously. This means that the interest is added to the account balance continuously throughout the year.
The formula for calculating the balance of an account with continuous compounding is:
[tex]V = Pe^(rt)[/tex]
where:
V = the balance after t years
P = the initial investment (or principal)
e = the mathematical constant approximately equal to [tex]2.71828[/tex]
r = the annual interest rate as a decimal
t = the number of years
Using this formula and substituting the given values, we get:
[tex]V = 479e^(0.08212)[/tex]
Simplifying this expression, we get:
[tex]V ≈ $1,204.70[/tex]
Therefore, the person's investment of [tex]$479[/tex] with an annual interest rate of [tex]8.2%[/tex] compounded continuously, would grow to approximately after 12 years
The formula for calculating the value of the account after t years, with continuous compounding, is:
[tex]V = Pe^(rt)[/tex]
where V is the final value, P is the initial principal, r is the interest rate (expressed as a decimal), and t is the time in years.
Using this formula, we can calculate the value of the account after 12 years:
[tex]V = 479 * 2.6709[/tex]
[tex]V = 1280.74[/tex]
Final answer
Therefore, the amount of money in the account after [tex]12[/tex] years, to the nearest cent, is [tex]$1,280.74.[/tex]
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