Arun is going to invest $7,700 and leave it in an account for 20 years. Assuming theinterest is compounded continuously, what interest rate, to the nearest hundredth ofa percent, would be required in order for Arun to endend up with $13,100?
So first you do 13,100 minus 7700 equals 5400. Now you have 5400 you want to divide it by 20 which equals 270. So over the course of 20 years it went up to 13100. Which means every year it had to go up by $270 but that’s not a percent so… we have to divide 13100 by 5400 which equals 2.43 (Hope this helps)
What is the relative maximum for f(x)=-x3+6x2-10x+4
The relative maximum of the function f(x)=-x³+6x²-10x+4 is 2.82 or 1.18.
What is the relative maximum of the function?
The relative maximum of the function f(x)=-x³+6x²-10x+4 is calculated as follows;
f(x)=-x³+6x²-10x+4
Take the first derivative of the function;
df(x)/dx = -3x² + 12x - 10
-3x² + 12x - 10 = 0
Solve the x in the quadratic equation above;
3x² - 12x + 10 = 0
Solve the equation using formula method;
x = 2.82 or 1.18
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11. Jon needs to order soccer
balls for his soccer team.
Each ball costs $24.99. The
shipping and handling costs
are $6.50. If he budgeted
$300, how many soccer
balls can he purchase?
Inequality:
Answer:
The amount of soccer balls that he would be able to purchase would be = 12 balls.
How to calculate the number of soccer balls that can be purchased?The cost of each ball = $24.99
The cost of shipping and handling costs = $6.50
The total amount that he budgeted = $300
The amount remaining after deduction of shipping cost = 300-6.50 = 293.5
The number of balls = 293.5/24.99
= 11.7 = 12 balls
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LOOK AT THE IMAGE AND ANSWER ASAP!!! FIRST TO DO IT CORRECTLY GETS BRAINLIEST!!!
Answer:
[tex]f(t) = - 16 {t}^{2} + 60t + 16[/tex]
A. x = (-60 + √(60^2 - 4(-16)(16))) / (2(-16))
= (-60 + √4,624)/-32
= (-60 + 68)/-32
= -1/4, 4
So the coordinates of the roots
(x-intercepts) are (-1/4, 0) and (4, 0).
B. The line of symmetry is
x = (-1/4 + 4)/2 = (15/4)(1/2) = 15/8 = 1.875
f(1.875) = -16(15/8)^2 + 60(15/8) + 16
= 72.25
So the vertex is (1.875, 72.25).
C. Plot the roots and the vertex on the graph. f(1) = 60, f(2) = 72, and f(3) = 52, so plot (1, 60), (2, 72), and (3, 52).
Then draw a smooth curve through all the points. The vertex of this graph is a maximum.
Which of the following is the graph of f(x)=√x-2+3?
Answer:
x y
0 1
1 2
2 2.41
Maria is solving this number riddle by using guess and check: Fifteen less than a number is twelve. Based on the riddle, which statements must be true? Select the four correct answers. Less than means subtraction. Subtract 15 from the unknown number to get 12. 15 minus 12 = 3 The unknown number is 27. 15 + 3 = 18 27 minus 12 = 15
The correct statement is,
⇒ The unknown number is 27.
We have to given that;
Maria is solving this number riddle by using guess and check:
⇒ Fifteen less than a number is twelve.\
Now, We can write as;
⇒ x - 15 = 12
Solve for x;
⇒ x = 15 + 12
⇒ x = 27
Thus, The correct statement is,
⇒ The unknown number is 27.
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A gardener would like to add to their existing garden to make more flowers available for the butterflies that visit the garden. Her current garden is 24 square feet. If she added another rectangular piece with vertices located at (−17, 15), (−20, 15), (−17, 11), and (−20, 11), what is the total area of the garden?
144 ft2
288 ft2
12 ft2
36 ft2
The total area of the garden is 36 ft2.
To find the area of the rectangular piece that the gardener wants to add to the existing garden, we need to find the length and width of the rectangle.
The length of the rectangle is the distance between the points (-17, 15) and (-20, 15), which is 3 units.
The width of the rectangle is the distance between the points (-17, 15) and (-17, 11), which is 4 units.
Therefore, the area of the rectangular piece that the gardener wants to add is 3 x 4 = 12 square feet.
To find the total area of the garden, we need to add the area of the existing garden to the area of the rectangular piece that the gardener wants to add:
24 + 12 = 36
Therefore, the total area of the garden will be 36 square feet.
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After making 20 servings of pasta a chef has used 30 cloves of garlic the shaft use 6 cloves to make the first four servings how many cloves of garlic are used to make 10 servings
Therefore, the chef would use 15 cloves of garlic to make 10 servings of pasta.
The chef used 30 cloves of garlic to make 20 servings of pasta. Therefore, the number of cloves of garlic used per serving is:
30 cloves / 20 servings = 1.5 cloves per serving
The chef used 6 cloves of garlic to make the first four servings of pasta. Therefore, the number of cloves of garlic used per serving for those four servings is:
6 cloves / 4 servings = 1.5 cloves per serving
So, the chef used 1.5 cloves of garlic per serving consistently. To find out how many cloves of garlic are used to make 10 servings, we can use a proportion:
1.5 cloves / 1 serving = x cloves / 10 servings
Cross-multiplying, we get:
1.5 cloves * 10 servings = x cloves * 1 serving
15 cloves = x cloves
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How do I get and solve for X?
The value of variable x is,
x = 7.77
We have to given that;
A triangle is shown in figure.
Now, We can formulate;
By using trigonometry identity,
tan 48° = x / 7
1.11 = x/7
x = 7 × 1.11
x = 7.77
Thus, The value of variable x is,
x = 7.77
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rewrite each proportion in fraction from. then find the value of each variable
×:8 = 9:24
The value of the variable is 3
What is proportion?Proportion can be defined as a method of comparing numbers in mathematics such that one is made equal to another.
Note that a fraction is described as the part of a whole
From the information given, we have that;
×:8 = 9:24
To determine the fraction, we divide the numerator by the denominator, we have;
x/8= 9/24
Now, cross multiply the values
24(x) =9(8)
multiply the values, we have;
24x = 72
Now, make 'x' the subject
Divide both sides by the coefficient
x = 3
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(Similar to an old final) Let X and Y be independent random variables with X = N(0,1) and Y = exp(1). Find E([[X|(Y+1)-1). Point/Hint: Use the concept in Lecture 13, slide 15. One of the most powerful
When X and Y be independent random variables with X = N(0,1) and Y = exp(1) then E([[X|(Y+1)-1)=0.
To find the expected value E([X|(Y+1)-1]), we first need to clarify the expression inside the brackets. Since Y+1-1 = Y, the expression becomes E([X|Y]). Now, let's proceed to find E(X|Y):
1. X and Y are independent random variables, with X following a normal distribution N(0, 1) and Y following an exponential distribution with a rate parameter of 1.
2. To find the expected value of X given Y, we can use the property of independent random variables:
E(X|Y) = E(X), since Y's value does not affect X's expected value X.
3. Given that X follows a normal distribution with a mean of 0 and a variance of 1, the expected value E(X) is equal to its mean, which is 0.
So, E([X|Y]) = E(X) = 0.
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true or false,Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.
Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy. This statement is True.
Finding an eigenvector of a matrix A involves solving the equation $(A - \lambda I)\vec{v} = \vec{0}$, where $\lambda$ is an eigenvalue of A and $\vec{v}$ is the corresponding eigenvector.
This can be a challenging computational problem in general, especially for larger matrices or complex eigenvalues.
However, once a candidate eigenvector is found, it is easy to check whether it is in fact an eigenvector.
Simply multiply the vector by A and compare the result to the product of the eigenvalue and the original vector.
If they are equal, then the vector is indeed an eigenvector. This verification process is straightforward and can be done quickly.
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Determine the product of 15/6 and 1.2
Answer:
3
Step-by-step explanation:
Let E be the region bounded below by the cone z = – (x2 + y²) and above by the sphere 102 – x2 - y2 . Provide an answer accurate to at least 4 significant digits. Find the volume of E.
The volume of E is 0.
We need to find the volume of the region E bounded below by the cone z = –(x^2 + y^2) and above by the sphere 102 – x^2 – y^2.
To find the limits of integration, we need to solve for z in terms of x and y:
z = –(x^2 + y^2)
z + x^2 + y^2 = 0
x^2 + y^2 = –z
And for the sphere:
102 – x^2 – y^2 = z
102 = x^2 + y^2 + z
Substituting x^2 + y^2 = –z in the equation for the sphere, we get:
102 = –z + z
102 = 0
This is impossible, so there is no intersection between the cone and the sphere, and the volume of E is zero.
Therefore, the volume of E is 0.
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Use the diagram to find the distances. Then write the distances to complete the equations.
plss help
The distances obtained using the distance formula indicates that the distances between the points are;
(A) QS = 17 units
(B) PS = 15 units
(C) SR = 6 units
(D) PQ = 17 units
(E) QR = 10 units
What is the distance formula?The distance formula is a formula that can be used to find the distance, d, between two points (x₁, y₁), and (x₂, y₂), on the coordinate plane, and can be presented as follows;
d = √((x₂ - x₁)² + (y₂ - y₁)²)
The corresponding values of the coordinate points indicates;
QS = 10 - 2 = 8 units
PS = 16 - 1 = 15 units
SR = 22 - 16 = 6 units
The Pythagorean Theorem and the distance formula indicates that we get;
PQ = √((16 - 1)² + (10 - 2)²) = 17 units
QR = √((22 - 16)² + (10 - 2)²) = 10 units
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Which of the following is a difference of cubes?
125^6 -9y^3
8x^3 - 27y^6
x^3+8y^3
3x^3-8y^6
The difference of cubes among the options is
8x^3 - 27y^6How to find the difference of cube
The equation of difference of cubes is given by:
a^3 - b^3 = (a - b) (a^2 + ab + b^2)
In this case, the difference of cubes is:
8x^3 - 27y^6
This can be rewritten as
= (2x)^3 - (3y^2)^3
here
a = 2x
b = 3y^2
Therefore, the correct answer is as follows:
8x^3 - 27y^6
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What function is a vertical shift of f(x) = x?
A) g(x) = 3f(x)
B) g(x) = f(x - 3)
C) g(x) = f(x) + 4
D) g(x) = 1/2 f(x)
Answer:
C) g(x) = f(x) + 4
Step-by-step explanation:
A vertical shift is where you shift, slide or translate the whole graph up or down (on a graph) The way this shows up in the equation is just a number tacked on to the end of the equation. A +anumber (like the +4 in the answer) slides the function UP four units. A
-anumber would slide the function DOWN instead.
As for the other answers:
A) the 3multiplied in front is a vertical STRETCH.
D) the 1/2 multiplied in front is a vertical shrink (smash)
B) The -3 in close tight with the x is a horizontal shift(slide, translate) It is a RIGHT shift. A +anumber would be a LEFT shift. Horizontal shift seem kind of backwards. + goes LEFT and - goes RIGHT.
Please answer this question for a quick 100 points
The weight of the candle after 5 hours of burning would be about 15.05 ounces.
If the burn rate is believed to be constant, we need to determine the average burn rate for the eight candles as the ratio of weight loss per hour.
Ounces lost over three hours =
0.5+0.6+0.5+0.7+0.7+0.5+0.5+0.6 = 4.6/8 = 0.575
Ounces lost per hour on average =
= 0.19
For 0 hours, the weight of each candle is 16 ounces.
Therefore, the equation can be =
w = 16 - 0.19h.
This model can be used to predict the weight of the candle when h, the number of hours of burning, is 5.
W = 16 - 0.19(5)
W = 16 - 0.95
W = 15.05
Hence the weight of the candle after 5 hours of burning would be about 15.05 ounces.
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A random sample of 100 customers at a local ice cream shop were asked what their favorite topping was. The following data was collected from the customers.
Topping Sprinkles Nuts Hot Fudge Chocolate Chips
Number of Customers 12 17 44 27
Which of the following graphs correctly displays the data?
a bar graph titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled sprinkles going to a value of 17, the second bar labeled nuts going to a value of 12, the third bar labeled hot fudge going to a value of 27, and the fourth bar labeled chocolate chips going to a value of 44
a bar graph titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled nuts going to a value of 17, the second bar labeled sprinkles going to a value of 12, the third bar labeled chocolate chips going to a value of 27, and the fourth bar labeled hot fudge going to a value of 44
a histogram titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled sprinkles going to a value of 17, the second bar labeled nuts going to a value of 12, the third bar labeled hot fudge going to a value of 27 ,and the fourth bar labeled chocolate chips going to a value of 44
a histogram titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled nuts going to a value of 17, the second bar labeled sprinkles going to a value of 12, the third bar labeled chocolate chips going to a value of 27, and the fourth bar labeled hot fudge going to a value of 44
Answer:
The correct graph that displays the collected data is: a bar graph titled favorite topping with the x axis labeled topping and the y axis labeled number of customers, with the first bar labeled Sprinkles going to a value of 12, the second bar labeled Nuts going to a value of 17, the third bar labeled Hot Fudge going to a value of 44, and the fourth bar labeled Chocolate Chips going to a value of 27.
This is the correct representation because a bar graph is used to display categorical data, and the x-axis represents the toppings while the y-axis represents the number of customers. The bars accurately show the frequency of each topping preference.
Regarding the second part of the prompt, there seems to be some missing information. Could you please provide the full question and options?
Step-by-step explanation:
9. Look at the graph below. If the object is rotated 180° about the z-axis, the coordinates for
Point A (-1, 2, 2) will be
1
The image of the point after it is rotated 180° about the z-axis is A' = (1, -2, -2)
Calculating the image of the point after the rotationFrom the question, we have the following parameters that can be used in our computation:
Point A = (-1, 2, 2)
The rule of rotation is given as rotated 180° about the z-axis
Mathematically, this rule can be expressed as
(x, y, z) = (-x, -y, -z)
Substitute the known values in the above equation, so, we have the following representation
A' = (1, -2, -2)
Hence. the image of the point after the rotation is A' = (1, -2, -2)
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Assume the random variable X is normally distributed, with mean and standard deviation . Find the percentile.Assume the random variable X is normally distributed, with mean μ=58 and standard deviation σ=8. Find the 11th percentile.
A percentile is a measure used in statistics to indicate the value below which a given percentage of observations fall. For example, if a data set has a 75th percentile value of 100, then 75% of the observations in the data set fall below the value of 100.
To find the 11th percentile for the given normal distribution with mean μ=58 and standard deviation σ=8, we need to use a standard normal distribution table or a calculator.
We can start by converting the given value to a z-score using the formula:
z = (X - μ) / σ
Where X is the value we want to find the percentile for, μ is the mean, and σ is the standard deviation.
Plugging in the values given, we get:
z = (X - 58) / 8
To find the z-score for the 11th percentile, we can use a standard normal distribution table or calculator to find the z-score associated with a cumulative probability of 0.11.
Using a calculator or table, we find that the z-score associated with a cumulative probability of 0.11 is -1.23.
We can now solve for X using the z-score formula:
z = (X - μ) / σ
-1.23 = (X - 58) / 8
Solving for X, we get:
X = -1.23 * 8 + 58 = 48.16
Therefore, the 11th percentile for this normal distribution is 48.16. This means that 11% of the observations in this distribution fall below the value of 48.16.
A Z-score represents the number of standard deviations an observation is from the mean of the distribution. The formula for calculating a Z-score is:
Z = (X - μ) / σ
In this case, you need to find the Z-score corresponding to the 11th percentile. To do this, you can refer to the Z-table, which provides the area (probability) to the left of a given Z-score. Look for the value closest to 0.11 (representing 11%) in the table. You will find that the Z-score associated with the 11th percentile is approximately -1.23.
Now, you can use the Z-score formula to solve for X:
-1.23 = (X - 58) / 8
To solve for X, perform the following calculations:
X - 58 = -1.23 * 8
X - 58 = -9.84
X = 58 - 9.84
X ≈ 48.16
So, the 11th percentile of the normally distributed random variable X with a mean of 58 and standard deviation of 8 is approximately 48.16.
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Aisha divides 1/5 meter of ribbon into 2 equal pieces
What is the length of each piece of ribbon
Enter your answer as a fraction in simplest form by filling in the boxes
The length of half Aisha's ribbon would be 1/10 meter
What is a fraction?A fraction can be defined as the part of a whole number, a whole element or variable.
The different types of fractions are;
Mixed fractionsProper fractionsImproper fractionsComplex fractionsSimple fractionsExample of mixed fractions; 2 1/3, 4 1/2, 41/3
Example of improper fractions are; 4/3, 5/4
Examples of simple fractions; 1/2, 1/4
From the information given, we have that;
The length of the ribbon is 1/5 meter
Half of the length would be;
1/5÷ 2
Divide the values
1/5 × 1/2
Multiply the values
1/10 meter
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find the value of y. give an exact answer.
Answer:
y = 8√2
Step-by-step explanation:
First, we can use the 30-60-90 triangle rule on the rightmost triangle. This states that:
the hypotenuse is twice the short legthe long leg is √3 times the short legSo, we can solve for the middle line (I will call it L) on this triangle:
L = 16 / 2
L = 8
Next, we can solve for y using the rule with 45-45-90 triangles:
the legs are congruentthe hypotenuse is √2 times the length of the legsSo, we can solve for y:
y = 8 · √2
y = 8√2
The volume of a spherical balloon is 3054 cm^3. Find the surface area of the balloon to the nearest whole number.
Select the buttons to rank the scenarios in order of most to least impact on your facility's overall results, Select here to rest bon You notice your inventory is lower than it should be for several popular Items Most Impact Select here to get button You overhear two associates complaining about a recent company-wide policy chango Least Impact here to reset button Your department needs to set up a display for an upcoming sales promotion that starts later this week 3rd Most Impact Select here to rostou You need to develop next week's work schedule for your associates 2nd Most impact
Developing a well-organized work schedule is crucial for the success of the facility and ensures that the business operations run smoothly, resulting in increased customer satisfaction, employee engagement, and profitability.
Developing the work schedule for the associates is the 2nd most impactful scenario as it directly affects the productivity and efficiency of the facility. A well-planned and balanced work schedule ensures that the right number of associates are scheduled at the right time to meet the demands of the business while minimizing unnecessary labor costs. Failure to create an effective schedule can result in understaffing, overstaffing, or scheduling conflicts, which can lead to a decrease in productivity and customer satisfaction.
The work schedule also affects employee satisfaction, as they need to balance their personal lives with their work schedules. A poorly managed schedule can lead to burnout, increased absenteeism, and high turnover rates. It is essential to consider associate availability, skill sets, and workload while developing the schedule to ensure that all tasks are completed efficiently and effectively.
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You may need to use the appropriate appendix table or technology to answer this question.
A group conducted a survey of 13,000 brides and grooms married in the United States and found that the average cost of a wedding is $21,858. Assume that the cost of a wedding is normally distributed with a mean of $21,858 and a standard deviation of $5,800.
(a)
What is the probability that a wedding costs less than $20,000? (Round your answer to four decimal places.)
(b)
What is the probability that a wedding costs between $20,000 and $31,000? (Round your answer to four decimal places.)
(c)
What is the minimum cost (in dollars) for a wedding to be included among the most expensive 5% of weddings? (Round your answer to the nearest dollar.)
$
The probability that a wedding costs less than $20,000 is approximately 0.3745.
The probability that a wedding costs between $20,000 and $31,000 is approximately 0.6188.
The minimum cost for a wedding to be included among the most expensive 5% of weddings is approximately $31,229.
(a) To find the probability that a wedding costs less than $20,000, we need to standardize the value of $20,000 by subtracting the mean and dividing by the standard deviation:
z = (20000 - 21858) / 5800 = -0.32
We can then use a standard normal distribution table or technology to find the corresponding probability:
P(z < -0.32) ≈ 0.3745
Therefore, the probability that a wedding costs less than $20,000 is approximately 0.3745.
(b) To find the probability that a wedding costs between $20,000 and $31,000, we need to standardize both values and find the area between the corresponding z-scores:
z1 = (20000 - 21858) / 5800 = -0.32
z2 = (31000 - 21858) / 5800 = 1.58
Using a standard normal distribution table or technology, we can find the probabilities:
P(-0.32 < z < 1.58) ≈ 0.6188
Therefore, the probability that a wedding costs between $20,000 and $31,000 is approximately 0.6188.
(c) To find the minimum cost for a wedding to be included among the most expensive 5% of weddings, we need to find the z-score that corresponds to the 95th percentile of the standard normal distribution. We can use a standard normal distribution table or technology to find this value:
z = invNorm(0.95) ≈ 1.645
We can then use the formula for standardizing a value to find the minimum cost:
z = (x - 21858) / 5800
Solving for x, we get:
x = z(5800) + 21858
x = 1.645(5800) + 21858
x ≈ 31229
Therefore, the minimum cost for a wedding to be included among the most expensive 5% of weddings is approximately $31,229.
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A shirt order consists of 10 small, 5 medium, and 8 large
shirts. The prices of the shirts are small $5.00; medium
$7.50; large $12.00. There is a mail order charge of $.50
per shirt for shipping and handling. Write an equation
for the total cost of ordering the shirts by mail.
The equation for the total cost of ordering the shirts by mail is Total = 10 * 5 + 5 * 7.5 + 8 * 12 + 0.50
Writing an equation for the total cost of ordering the shirts by mail.From the question, we have the following parameters that can be used in our computation:
Order = 10 small, 5 medium, and 8 largePrices = small $5.00; medium $7.50; large $12.00. Mail order charge of $.50The total cost of ordering the shirts by mail is then represented as
Total = Sum of price * order + Mail order charge
Using the above as a guide, we have the following:
Total = 10 * 5 + 5 * 7.5 + 8 * 12 + 0.50
When evaluated, we have
Total = 184
Hence, the total amount of order is $184
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Given that W is the center of the circle and that TS = VU, Find VU if WX= 4 and WS = 6. Round the answer to two decimal places. A. 4.47 B. 7.07 C. 8.94 D. 9.13
In the given circle, by using the Pythagorean theorem, the value of VU is 8.94
Calculating the value of VU in the circleFrom the question, we are to determine the value of VU in the given circle
From the given information, we have that
WX = 4
and
WS = 6
From the Pythagorean theorem, we can write that
|WS|² = |WX|² + |XS|²
6² = 4² + |XS|²
36 = 16 + |XS|²
36 - 16 = |XS|²
20 = |XS|²
|XS| = √20
In the given diagram, WX bisects chord TS.
Thus,
We can write that
|TS| = 2 × |XS|
|TS| = 2 × √20
|TS| = 2√20
|TS| = 8.94
From the given information, we were given that
TS = VU
Hence,
VU = 8.94
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Which of the following is a statistical question?
Responses
How many letters are in my name?
How many letters are in my name?
What is my favorite subject in school?
What is my favorite subject in school?
How many televisions are in my house?
How many televisions are in my house?
What are the heights of the students in my history class?
What are the heights of the students in my history class?
The statistical question is what are the heights of the students in my history class?
What is a statistical question?A statistical question is a question that can be answered by collecting data that vary. For example, the heights of students in your class would be different. Some students would be really tall while others would be short.
The number of letters in your name is constant. For example, if your name is Amy. There would be always be three letters in your name. Thus, it is not a statistical question.
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note: this is a multi-part question. once an answer is submitted, you will be unable to return to this part. what is the probability that bo, colleen, jeff, and rohini win the first, second, third, and fourth prizes, respectively, in a drawing if 50 people enter a contest and satisfying the following conditions? (enter the value of probability in decimals. round the answer to two decimal places.) winning more than one prize is allowed.
To find the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in a drawing with 50 people, follow these steps:
1. Since winning more than one prize is allowed, the probability of Bo winning the first prize is 1/50.
2. Likewise, the probability of Colleen winning the second prize is also 1/50.
3. Similarly, the probability of Jeff winning the third prize is 1/50.
4. Finally, the probability of Rohini winning the fourth prize is 1/50.
5. Since these events are independent, we can multiply the probabilities together to find the overall probability of this specific :
Probability = (1/50) * (1/50) * (1/50) * (1/50)
6. Calculate the result:
Probability ≈ 0.00000016
7. Round the answer to two decimal places:
Probability ≈ 0.00
So, the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in a drawing with 50 people is approximately 0.00.
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Redd is an art dealer, and is loading paintings into boxes for transportation. Each of Redd's paintings are unique and different from all the other paintings; however, the packaging boxes are identical. (a) How many ways are there to place 10 paintings into 8 boxes, given there should be at least one painting in each box? (b) How many ways are there to place 5 paintings into 3 boxes, if we are allowed to leave some boxes empty.
The number of ways to place 10 paintings into 8 boxes with at least one painting in each box is C(9,2) = 36.
The number of ways to place 5 paintings into 3 boxes when some boxes can be empty is C(7,2) = 21.
(a) To place 10 paintings into 8 boxes with at least one painting in each box, we can use the concept of distributing identical items into distinct groups. We will first place one painting in each box, leaving us with 2 paintings to distribute among the 8 boxes. We can use the "stars and bars" method to solve this problem. We have 2 "stars" (paintings) and need to separate them using 7 "bars" (box dividers). We can think of this as choosing 2 positions from 9 available positions (2 stars + 7 bars). Therefore, the number of ways to place 10 paintings into 8 boxes with at least one painting in each box is C(9,2) = 36.
(b) To place 5 paintings into 3 boxes without the restriction that each box must have a painting, we will again use the "stars and bars" method. In this case, we have 5 "stars" (paintings) and 2 "bars" (box dividers). We can think of this as choosing 2 positions from 7 available positions (5 stars + 2 bars). Therefore, the number of ways to place 5 paintings into 3 boxes when some boxes can be empty is C(7,2) = 21.
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