A sample size a 28 produced test statistic is T equals 2. 51

Manuel can type 68 words in two minutes at a rate of 34 words per minute.

Given that; Manuel types at a rate of 34 words per minute.

To determine how many words Manuel can type in two minutes, we simply need to multiply his typing rate by the number of minutes he is typing.

Since Manuel is typing for two minutes

Hence;

Number of words = Typing rate × Time

Plugging in the values we have from the problem.

Number of words = 34 words/minute × 2 minutes

Simplifying

Number of words = 34 words × 2

Number of words = 68 words

Therefore, he can type 68 words in two minutes.

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When multiplying a whole number with a fraction, the first factor will be reduced to a fraction or a decimal, depending on the problem.

In both of the given problems, we are multiplying a whole number with a fraction. When we multiply a whole number with a fraction, we can simply multiply the whole number with the numerator of the fraction and keep the denominator as it is. So, in the first problem, we have 10 multiplied by 1/2. Multiplying 10 with 1 gives us 10, and then we divide it by 2, which gives us 5. Therefore, the first factor, which is 10, will be reduced to 5 in this problem.

In the second problem, we have 15 multiplied by 7/2. Multiplying 15 with 7 gives us 105, and then we divide it by 2, which gives us 52.5. Therefore, the first factor, which is 15, will be reduced to 52.5 in this problem.

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The table that have a constant of proportionality between y and x of 12 is the first table

From the question, we have the following parameters that can be used in our computation:

The table of values

From the first table of values, we have the following readings

(x, y) = (1/2, 6), (2, 24) and (10, 120)

Using the above as a guide, we have the following:

The constant of proportionality between y and x in the graph is

k = y/x

Substitute the known values in the above equation, so, we have the following representation

k = 6/(1/2) = 24/2 = 120/10

Evaluate

k = 12 = 12 = 12

Hence, the constant of proportionality between y and x in the first table is 12

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Complete question

Which table has a constant of proportionality between y and x of 12?

x          1/2               2             10

y           6                24          120

x          1/4               3             12

y           3                60          144

x          1/3               6             9

y           4                78          117

24 cm  is the height of cone .

What is known as a cone?

A cone is a three-dimensional geometric object with a smooth transition from a flat, generally circular base to the apex, also known as the vertex.

                                      A cone is a three-dimensional geometric structure with a smooth transition from a flat base—often but not always circular—to the point at the top, also known as the apex or vertex. Cone. a right circular cone having the following measurements: height, slant height, angle, base radius, and height.

V=1/3hπr²

V = 1/3 * h * 12π

96π = 1/3 * h * 12π

96π * 3/12π  = h

 8 * 3   = h

  h = 24 cm

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In a circle with radius 6 and angle intercepts an arc of length 3π , the angle in radians in simplest form is π/2.

In a circle, the length of an arc is proportional to the angle that it intercepts. The ratio of the arc length to the circumference of the circle is equal to the ratio of the angle in radians to 2π. Thus, we can write:

(arc length) / (circumference) = (angle) / (2π)

In this problem, we are given that the circle has a radius of 6 and that the arc length is 3π. We can use the formula for the circumference of a circle, which is C = 2πr, to find the circumference of this circle:

C = 2πr = 2π(6) = 12π

Now we can use the formula above to find the angle in radians:

(3π) / (12π) = (angle) / (2π)

Simplifying this equation, we get:

angle = (3π * 2π) / 12π = 1/2 * π

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If a rectangular piece of cardboard with dimensions 5 inches by 8 inches is used to make the curved side of a cylinder-shaped container, the greatest volume the cylinder can hold is 80/π cubic inches.

To find the greatest volume the cylinder can hold, we need to determine the dimensions of the cylinder that can be made from the given cardboard.

First, we need to calculate the circumference of the cylinder using the length of the cardboard, which will be the height of the cylinder. The length of the cardboard is 8 inches, so the circumference of the cylinder will be 8 inches.

The circumference of a cylinder is given by the formula C = 2πr, where r is the radius of the cylinder.

Therefore, 8 = 2πr, or r = 4/π inches.

Next, we need to determine the length of the curved side of the cylinder, which is given by the formula L = 2πr.

So, L = 2π(4/π) = 8 inches.

Finally, we can calculate the volume of the cylinder using the formula V = πr²h, where h is the height of the cylinder, which is 5 inches.

V = π(4/π)²(5) = 80/π cubic inches.

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The system of equation is 7.71 feet, under the condition the area of a rug is 108 square feet and the length it it’s diagonal is 14 feet.

Now to solve this problem, we can use the formula for the area of a rectangle which is A = L x W . Therefore, we can write the equation 108 = L x W

Now the length of the diagonal is 14 feet. We can use this information to write another equation using the Pythagorean theorem which states that for any right triangle with legs of length a and  b  and hypotenuse of length c ,
a² + b² = c²
Since a rectangle is made up of two right triangles, we can use this theorem to find the length and width of the rectangle.

Let us assume the length of the rug L and the width of the rug  W

L²+ W² = 14²

We have two equations with two unknowns

108 = L x W
L² + W² = 14²

We can solve for one variable in terms of another using substitution. From the first equation,

W = 108 / L

Substituting this into the second equation gives:


L² + (108 / L)² = 14²
L² - 196L² + 11664 = 0


This is a quadratic equation in terms of  L². We can solve for L²  


L² = (196 ± √(196² - 4 x 11664)) / 2
L² = (196 ± √(38416)) / 2
L² = (196 ± 196) / 2


Taking the positive root gives:

L² = 196

So:

L = √(196) = 14

Substituting this back into one of our original equations gives:

W = 108 / L
= 108 / 14
≈ 7.71

Therefore, the length of the rug is 14 feet and its width is approximately 7.71 feet.
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The solution to the system of differential equations x' = y - x + t and y' = y with initial conditions x(0) = 9 and y(0) = 4 is x(t) = 10e^t - t - 1 and y(t) = 9e^t - 5t - 5.To find this solution, we first solve for y in the second equation:y' - y = 0y(t) = Ce^tNext, we substitute this expression for y into the first equation and solve for x:x' = Ce^t - x + tx' + x = Ce^t + tMultiplying both sides by e^t, we get:(e^t x)' = Ce^2t + te^tIntegrating both sides:e^t x(t) = (C/2)e^2t + te^t + DUsing the initial condition x(0) = 9, we get:D = 9Using the expression for y(t) and the initial condition y(0) = 4, we get:C = 5Substituting these values into the equation for x(t), we get:x(t) = 10e^t - t - 1Finally, we substitute the expression for y(t) into the given initial condition y(0) = 4 and solve for the constant C:C = 9 - 5tSubstituting this expression for C into the equation for y(t), we get:y(t) = 9e^t - 5t - 5

For more similar questions on topic Vectors in 2D is a sub-topic in linear algebra that deals with the study of vectors in two-dimensional space. In two-dimensional space, vectors are represented as ordered pairs of real numbers and can be used to describe quantities such as displacement, velocity, and force. The magnitude and direction of a vector can be calculated using trigonometry, and vectors can be added, subtracted, and multiplied by scalars using the rules of vector algebra.

In the context of the given problem, we are asked to find two unit vectors in 2D that make an angle of 45 degrees with a given vector 6i + 5j, where i and j are the unit vectors in the x and y directions, respectively. To solve this problem, we need to use the properties of vectors and trigonometry to find the appropriate unit vectors that satisfy the given conditions. The solution to this problem involves finding the components of the given vector, calculating the angle between this vector and the x-axis, and using this angle to construct the desired unit vectors.

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The solution to the system of differential equations is:
x(t) = 5 e^(t/2) - 4 e^(3t/2)
y(t) = 4

To solve this system of differential equations, we can use Laplace transforms. Taking the Laplace transform of both sides of each equation, we get:

sX(s) - x(0) = Y(s) - X(s) + T Y(s)
sY(s) - y(0) = Y(s)

Substituting in the initial conditions x(0) = 9 and y(0) = 4, we can solve for X(s) and Y(s):

X(s) = (s + 1)/(s^2 - s - T)
Y(s) = 4/s

To find x(t) and y(t), we need to inverse Laplace transform these expressions. We can use partial fractions to simplify the expression for X(s):

X(s) = A/(s - r1) + B/(s - r2)

where r1 and r2 are the roots of the denominator s^2 - s - T, given by:

r1 = (1 - sqrt(1 + 4T))/2
r2 = (1 + sqrt(1 + 4T))/2

Solving for A and B, we get:

A = (r2 + 1)/(r2 - r1)
B = -(r1 + 1)/(r2 - r1)

Substituting these values back into the expression for X(s), we get:

X(s) = (r2 + 1)/(r2 - r1)/(s - r1) - (r1 + 1)/(r2 - r1)/(s - r2)

Taking the inverse Laplace transform of this expression, we get:

x(t) = (r2 + 1)/(r2 - r1) e^(r1 t) - (r1 + 1)/(r2 - r1) e^(r2 t)

Substituting in the values for r1 and r2, we get:

x(t) = 5 e^(t/2) - 4 e^(3t/2)

Similarly, taking the inverse Laplace transform of Y(s) = 4/s, we get:

y(t) = 4

Therefore, the solution to the system of differential equations is:

x(t) = 5 e^(t/2) - 4 e^(3t/2)
y(t) = 4

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The unit rate of the linear equation y = 8x is 8.

A linear equation of a function illustrates the straight line on a coordinate plane. It can be expressed in a slope intercept form y = mx + b, where:

The slope shows steep the gradient is and it is the change in the rise over the run. For a unit rate which is usually the ratio between two different quantities. We can say in the linear equation, the unit rate represents the slope of the function.

Given that:
y = 8x

To find the unit rate(slope) which is the change in the y-axis(rise) over the change in the x-axis(run) by using slope-intercept form. Then, we can conclude that the unit rate is 8.

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Answer:

ChatGPT

To find the balances for each type of investment after one year, we can use the formula for compound interest:

Treasury bond: A = P(1 + r/n)^(nt)

A = 0.4(30000)(1 + 0.0535/1)^(1*1) = $12,912.00

CD: A = P(1 + r/n)^(nt)

A = 0.1(30000)(1 + 0.0475/1)^(1*1) = $10,316.25

Stock plan: After the first year, 30% is still in the savings account. The other 70% is in the stock plan, which increased by 9%, so the new value is:

0.7(30000)(1 + 0.09) = $23,940.00

Savings account: A = P(1 + r/n)^(nt)

A = 0.2(30000)(1 + 0.039/1)^(1*1) = $6,351.00

To find the total gain from all of the investments combined, we need to add up the gains from each investment:

Treasury bond: $12,912.00 - $12,000.00 = $912.00 gain

CD: $10,316.25 - $9,000.00 = $1,316.25 gain

Stock plan: After the second year, the stock plan decreased in value by 5%, so the new value is:

0.7($23,940.00)(1 - 0.05) = $19,149.00

After the third year, the stock plan increased by 7%, so the final value is:

0.7($19,149.00)(1 + 0.07) = $20,129.57

The gain from the stock plan is:

$20,129.57 - $21,000.00 = -$870.43 loss (since the stock plan decreased in value overall)

Savings account: $6,351.00 - $6,000.00 = $351.00 gain

Total gain = $912.00 + $1,316.25 - $870.43 + $351.00 = $708.82

If you had invested 40% in stock and 30% in Treasury bonds, the calculations would be:

Treasury bond: A = P(1 + r/n)^(nt)

A = 0.3(30000)(1 + 0.0535/1)^(1*3) = $12,853.81

Stock plan: After the first year, 40% is still in the savings account. The other 60% is in the stock plan, which increased by 9%, so the new value is:

0.6(30000)(1 + 0.09) = $16,200.00

After the second year, the stock plan decreased in value by 5%, so the new value is:

0.6($16,200.00)(1 - 0.05) = $15,390.00

After the third year, the stock plan increased by 7%, so the final value is:

0.6($15,390.00)(1 + 0.07) = $16,019.16

Total gain = ($12,853.81 - $12,000.00) + (-$981.84) + ($1,019.16) = $890.13

Therefore, investing 40% in stock and 30% in Treasury bonds

Answer: 35 + 25 + 50 / 2 = 85

Step-by-step explanation: You would have to add them all together and then divide them by 2.

Answer:

0 hrs 32 mins

Step-by-step explanation:

The probability of  P(Ski) is 7 / 30, P(Snowboard) is 13 / 30,P(Ski & Snowboard) is 2/15 and P(ski or snowboard) is  8/15.

1. Probability of a student skiing (P(Ski)):
  P(Ski) = number of students who ski / total number of students = 28 / 120 = 7 / 30

2. Probability of a student snowboarding (P(Snowboard)):
  P(Snowboard) = number of students who snowboard / total number of students = 52 / 120 = 13 / 30

3. Probability of a student skiing and snowboarding (P(Ski & Snowboard)):
  P(Ski & Snowboard) = number of students who ski and snowboard / total number of students = 16 / 120 = 4 / 30

=2/15

4.Probability(ski or snowboard) = (7/30) + (13/30) - (2/15)

P(ski or snowboard) = 8/15
Therefore, the probabilities are:

P(ski) = 7/30

P(snowboard) = 13/30

P(ski and snowboard) = 2/15

P(ski or snowboard) = 8/15

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Diego selling raffle tickets for $1.75 per ticket and she earned for 50 tickets is $87.50.

When a purchase, appropriation, requisition, or direct engagement with the customer occurs at the point of sale, the seller or supplier of the products or services completes a transaction. Title (property or ownership) of the object is transferred, and a price is settled, meaning a price is agreed upon for which the ownership of the item will transfer.

We can calculate how much money Diego would make if he sold each quantity of raffle tickets for $1.75 each using Excel's multiplication function. It is possible to create a table with the number of tickets sold in one column and the money taken in the other.

Diego would receive $17.50, for instance, if he sold 10 tickets (10 x $1.75). If he sold 20 tickets, he would earn $35 (20 x $1.75), and so on. Using Excel's fill handle, you can quickly fill the table with the totals for each sold ticket.

The table would look like this:

Number of Tickets Sold | Amount of Money Earned

•----------------------------------|---------------------------------------•

   10                                |    $17.50

   20                               |    $35.00

   30                               |    $52.50

   40                               |    $70.00

   50                               |    $87.50

By using the multiplication function in Excel, we can quickly calculate the amount of money Diego would earn for any number of raffle tickets sold at $1.75 per ticket.

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Check the picture below.

Answer:

C (12)

Step-by-step explanation:

The explicit equation for the nth term of the arithmetic sequence is an = 9 - 2.7n.

What is the implicit equation?

An implicit equation is an equation in which the variables are not explicitly expressed in terms of each other. In other words, the equation does not give a direct formula for one of the variables in terms of the other(s), but rather relates the variables through some function or equation.

What is the explicit equation?

An explicit equation is an equation in which one variable is expressed directly in terms of the other(s). In other words, the equation gives a formula for one of the variables in terms of the other(s).

According to the given information:

the first term of the sequence is a1 = 6.3, and the common difference between consecutive terms is d = -2.7 (since we subtract 2.7 from each term to get to the next term). Therefore, the explicit equation for the nth term of the sequence is:

an = 6.3 + (n - 1)(-2.7)

Simplifying this expression, we get:

an = 6.3 - 2.7n + 2.7

an = 9 - 2.7n

So the correct equation for the nth term of the arithmetic sequence 6.3, 3.6, 0.9, -1.8, -4.5, ... is:

an = 9 - 2.7n

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Answer:20 feet by 10 feet

Step-by-step explanation:

1)  The domain of Y = 3x⁴ + 4x³ is (-∞, ∞).

2) The x-intercepts are (0, 0) and (-4/3, 0) and the y-intercept is (0, 0)

3)  The horizontal asymptote is y = infinity.

4) Function does not exhibit any symmetry with respect to the y-axis or origin.

5)  The critical points are x = 0 and x = -1.

6)  The critical points are x = 0 and x = -1.

7) The function is concave down on the interval (-∞, -2/3) and concave up on the intervals (-2/3, 0) and (0, ∞).

1) The domain of a polynomial function is all real numbers, so the domain of Y = 3x⁴ + 4x³ is (-∞, ∞).

2) To find the x-intercepts, we set Y equal to zero and solve for x:

0 = 3x⁴ + 4x³

0 = x³(3x + 4)

x = 0 or x = -4/3

Therefore, the x-intercepts are (0, 0) and (-4/3, 0).

To find the y-intercept, we set x equal to zero and solve for Y:

Y = 3(0)⁴ + 4(0)³

Y = 0

Therefore, the y-intercept is (0, 0).

3) Polynomial functions do not have vertical asymptotes. However, as x approaches positive or negative infinity, the function approaches infinity. Therefore, the horizontal asymptote is y = infinity.

4) The function Y = 3x⁴ + 4x³ is neither even nor odd. Therefore, it does not exhibit any symmetry with respect to the y-axis or origin.

5) To find the critical points, we take the first derivative of Y and set it equal to zero:

Y' = 12x³ + 12x²

0 = 12x²(x + 1)

Therefore, the critical points are x = 0 and x = -1.

6) To determine whether the critical points are maxima or minima, we take the second derivative of Y and evaluate it at each critical point:

Y'' = 36x² + 24x

At x = 0, Y'' = 0, which means that the second derivative test is inconclusive. To determine whether x = 0 is a maxima or minima, we look at the sign of the first derivative to the left and right of the critical point. We find that Y' is negative to the left of x = 0 and positive to the right, so x = 0 is a local minimum.

At x = -1, Y'' = 12, which is positive. Therefore, x = -1 is a local minimum.

7) To determine the concavity of the function, we look at the sign of the second derivative:

Y'' = 36x² + 24x

When Y'' > 0, the function is concave up, and when Y'' < 0, the function is concave down.

At x < -2/3, Y'' is negative, so the function is concave down.

At -2/3 < x < 0, Y'' is positive, so the function is concave up.

At x > 0, Y'' is positive, so the function is concave up.

Therefore, the function is concave down on the interval (-∞, -2/3) and concave up on the intervals (-2/3, 0) and (0, ∞).

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The values of a, b, and c that represent the perimeter of the pool are a = 80, b = 0, and c = 100.

Step 1: Add the three dimensions together to find the total length of one side of the perimeter:
(4x^2 + 15) + (8x^2 + 10) + (8x^2) = 20x^2 + 25

Step 2: Since the perimeter has 4 equal sides (it's a rectangle), multiply the total length of one side by 4:
Perimeter = 4(20x^2 + 25) = 80x^2 + 100

Now, compare the perimeter polynomial with the general form ax^2 + bx + c:
80x^2 + 100 = ax^2 + bx + c

From this comparison, you can see that:
a = 80
b = 0 (since there is no term with x)
c = 100

So, the values of a, b, and c that represent the perimeter of the pool are a = 80, b = 0, and c = 100.

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The area of the table is approximately 651 square inches.

What is Area ?

Area is a measure of the size of a two-dimensional shape or surface, such as a rectangle, circle, or triangle. It is expressed in square units, such as square inches, square feet, or square meters.

Let's use the Pythagorean theorem to find the length of the table top:

Substituting the given values, we get:

40*40 = [tex]length^{2}[/tex] + 21*21

Simplifying and solving for length, we get:

[tex]length^{2}[/tex]= 1600 - 441

[tex]length^{2}[/tex] = 961

length = 31 inches (rounded to the nearest inch)

Now that we know the length and width of the table, we can find the area by multiplying them together:

area = length x width

area = 31 x 21

area = 651 square inches (rounded to the nearest inch)

Therefore, the area of the table is approximately 651 square inches.

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we can see that there is no stem value of 4 and therefore no department store with 49 pairs of glasses.

To find the largest number of pairs of glasses at any one department store, we need to examine the stem-and-leaf plot provided.

The stem-and-leaf plot shows the number of pairs of glasses at each store, with the first digit (the stem) indicating the tens place and the second digit (the leaf) indicating the ones place.

Looking at the plot, we can see that the largest stem is 4, which corresponds to the number 40. The largest leaf for stem 4 is 8, which corresponds to the number 48. Therefore, the largest number of pairs of glasses at any one department store is 48.

We can also verify this by scanning through the leaves in the plot and looking for the largest value. The largest leaf value is 9, which corresponds to the number 49. However, we can see that there is no stem value of 4 and therefore no department store with 49 pairs of glasses.

The largest number of pairs of glasses at any one department store is indeed 48.

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60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 19.1 and 37.5 HCF.

The correct compound inequality to represent the scenario is:

60 ≤ 1.36x + 34 ≤ 85

To solve for x, we need to isolate it in the middle of the inequality:

60 - 34 ≤ 1.36x ≤ 85 - 34

26 ≤ 1.36x ≤ 51

Finally, we divide by 1.36 to isolate x:

19.12 ≤ x ≤ 37.5

Therefore, the recommended range of water consumption is between 19.1 and 37.5 HCF. The answer is (D) 60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 19.1 and 37.5 HCF.

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complete question

To conserve water, many communities have developed water restrictions. The water utility charges a fee of $34, plus an additional $1.36 per hundred cubic feet (HCF) of water. The recommended monthly bill for a household is between $60 and $85 dollars per month. If x represents the water usage in HCF in a household, write a compound inequality to represent the scenario and then determine the recommended range of water consumption. (Round your answer to one decimal place.)

60 ≤ 1.36x − 34 ≤ 85; To stay within the range, the usage should be between 69.1 and 87.5 HCF.

60 ≤ 1.36x − 34 ≤ 85; To stay within the range, the usage should be between 44.1 and 87.5 HCF.

60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 37.5 and 44.1 HCF.

60 ≤ 1.36x + 34 ≤ 85; To stay within the range, the usage should be between 19.1 and 37.5 HCF.

Part A: The equation for the amount of air still needed is: y = 14,137.17 - 1600X

Part B: The total amount of air still needed to fill the ball after 4 minutes is 8,937.17 cubic inches.

Part C: It takes approximately 8.84 minutes to pump up the ball to its maximum volume.

Part A:

The formula for the volume of a sphere is 4/3πr³, where r is the radius. Since the maximum diameter of the exercise ball is 30 inches, its radius is 15 inches. Therefore, the maximum volume of the ball is:

4/3π(15)³ = 14,137.17 cubic inches

Let's let y represent the amount of air still needed to fill the ball to its maximum volume, and X represent the number of minutes the pump has been running. We know that the pump is filling the ball at a rate of 1600 cubic inches per minute. Therefore, the equation for the amount of air still needed is:

y = 14,137.17 - 1600X

Part B:

After 4 minutes, the pump has filled the ball with:

1600 x 4 = 6400 cubic inches

Using the equation from Part A, we can find the amount of air still needed after 4 minutes:

y = 14,137.17 - 1600(4) = 8,937.17 cubic inches

Therefore, the total amount of air still needed to fill the ball after 4 minutes is 8,937.17 cubic inches.

Part C:

To find the estimated number of minutes it takes to pump up the ball to its maximum volume, we can set the equation from Part A equal to 0 (since y represents the amount of air still needed):

0 = 14,137.17 - 1600X

Solving for X, we get:

X = 8.84

Therefore, it takes approximately 8.84 minutes to pump up the ball to its maximum volume.

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The following statements are true:

mabd + mdbc = 180 degrees

mabd = mevc

According to the definition of linear pairs, the two angles add up to 180 degrees. Therefore, mabd + mdbc = 180 degrees.

According to the definition of vertical angles, they have the same measure. Therefore, mabd = mevc.

Since abd and evc are vertical angles, and mabd = mevc, then mabd = mebc. Therefore, we can substitute mabd in the equation mebc = y+135/2 to get 5(2x + 1) = y+135/2.

We can solve for y to get y = 10x + 260.

Now we can substitute this value of y into the equation mebc = y+135/2 to get mebc = 10x + 347.5.

Therefore, none of the statements in the question that mention mebc are necessarily true or false, since we don't have enough information about the value of x to determine its measure.

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To evaluate the given integral using cylindrical coordinates, we need to first express the given solid E and the differential volume element dv in terms of cylindrical coordinates.

In cylindrical coordinates, the paraboloid z = 1 – x^2 - y^2 can be expressed as z = 1 – r^2, where r is the distance from the z-axis and θ is the angle made with the positive x-axis. Since the solid E lies in the first octant, we have 0 ≤ r ≤ √(1-z), 0 ≤ θ ≤ π/2, and 0 ≤ z ≤ 1 – r^2.

The differential volume element dv in cylindrical coordinates is given by dv = r dz dr dθ.

Substituting these expressions in the given integral, we get:

SITE . 742 + x^2 dv = ∫∫∫E (742 + r^2) r dz dr dθ

= ∫θ=0π/2 ∫r=0√(1-z) ∫z=0^(1-r^2) (742 + r^2) r dz dr dθ

= ∫θ=0π/2 ∫r=0√(1-z) [(742r + r^3/3) - (742r^3/3 + r^5/5)] dr dθ

= ∫θ=0π/2 ∫z=0^1 [247/3(1-z)^(3/2) - 185/6(1-z)^(5/2)] dz dθ

= ∫θ=0π/2 [98/15 - 185/21] dθ

= ∫θ=0π/2 [56/315] dθ

= [28/315]π

Therefore, the value of the given integral using cylindrical coordinates is [28/315]π.

To evaluate the given integral using cylindrical coordinates, we need to express the function and limits of integration in terms of cylindrical coordinates (r, θ, z). The conversion between Cartesian and cylindrical coordinates is given by:

x = r*cos(θ)
y = r*sin(θ)
z = z

The given function in the problem is z = 1 - x^2 - y^2. Substituting the expressions for x and y in terms of cylindrical coordinates, we get:

z = 1 - r^2(cos^2(θ) + sin^2(θ))
z = 1 - r^2

Now, we need to find the limits of integration for r, θ, and z. Since E is the solid in the first octant, the limits for θ are 0 to π/2. For r, the limits are 0 to √(1 - z), and for z, the limits are 0 to 1. Then, the integral becomes:

∫(0 to π/2) ∫(0 to √(1 - z)) ∫(0 to 1) (742 + r^2cos^2(θ) + r^2sin^2(θ)) * r dz dr dθ

Solve this triple integral to find the volume of the solid E.

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There are different ways a person can make up a story about a word problem. The square root of 81 - 7 is 2. Hence the story is given below.

The hypothetical story goes like this: Mia was trying to figure out the length of the missing side of a square garden bed. She knew that the total area of the garden bed was 81 square feet. She also knew that she had already planted in a section of the garden bed that was 7 feet wide.

So, what will be the length of the missing side of the garden bed if Mia wants to make sure it is completely filled with soil. In order to solve this problem, Mia used the equation: the square root of 81-7=2. She plugged in the values she knew and solved for the missing side length. Therefore, The solution will be:= ✓81 - 7= 9 - 7= 2

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A story for the word problem is that what will be the length of the missing side of a square garden when the total area is 81 feet² and the width was 7 feet wide?

From the information, we are seeking a story problem that has the equation had the square root of 81 - 7 = 2.

This will be written as story for the word problem is that what will be the length of the missing side of a square garden when the total area is 81 feet ² and the width was 7 feet wide?

The length will be calculated thus :

= √ 81 - 7

= 9 - 7

= 2

The length of the missing side is 2 feet.

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The instant when the radar gun is used, the rate at which the distance between the two cars is changing is g'(t) = 7968t + 368/5 kilometers per hour.

Let's break down the problem. We have two cars, one traveling south at 112 km/h and another traveling west at 96 km/h. The police car is stationed at an intersection and the two cars are at different points relative to the intersection. The first car is 4/5 kilometer north of the intersection while the second car is 2/5 kilometer east of the intersection.

Let's call this distance "d". Using the Pythagorean theorem, we can write:

d² = (4/5)² + (2/5)² d² = 16/25 + 4/25 d² = 20/25 d = sqrt(20)/5 d = 2sqrt(5)/5 kilometers

Now, we need to find the rate at which the distance between the two cars is changing. This is equivalent to finding the derivative of the distance with respect to time. Let's call this rate "r".

To find "r", we need to use the chain rule. The distance between the two cars is a function of time, so we can write:

d = f(t)

where t is time. We can then write:

r = d'(t) = f'(t)

where d'(t) and f'(t) denote the derivatives of d and f with respect to time, respectively.

To find f'(t), we need to express d in terms of t. We know that the first car is traveling at a constant speed of 112 km/h due south. Let's call the position of the first car "x" and the time "t". Then we have:

x = -112t

The negative sign indicates that the car is moving south. Similarly, we can express the position of the second car in terms of time. Let's call the position of the second car "y". Then we have:

y = 96t

The positive sign indicates that the car is moving west.

Now, we can use these expressions to find the distance between the two cars as a function of time. Let's call this function "g(t)". Then we have:

g(t) = √((x + 4/5)² + (y - 2/5)²) g(t) = √((-112t + 4/5)² + (96t - 2/5)²)

To find g'(t), we need to use the chain rule. We have:

g'(t) = (1/2)(x + 4/5)'(x + 4/5)'' + (y - 2/5)'x(y - 2/5)''

where the primes denote derivatives with respect to time. We can simplify this expression by noting that x' = -112 and y' = 96. We also have x'' = y'' = 0, since the speeds of the two cars are constant.

Substituting these values, we get:

g'(t) = -112x(-112t + 4/5)/√((-112t + 4/5)² + (96t - 2/5)²) + 96x(96t - 2/5)/√((-112t + 4/5)² + (96t - 2/5)²)

Simplifying this expression, we get:

g'(t) = (-112x(-112t + 4/5) + 96x(96t - 2/5))/√((-112t + 4/5)² + (96t - 2/5)²)

We can further simplify this expression by multiplying out the terms in the numerator:

g'(t) = (-12544t + 560/5 + 9216t - 192/5)/√((-112t + 4/5)² + (96t - 2/5)²)

g'(t) = (7968t + 368/5)/√((-112t + 4/5)² + (96t - 2/5)²)

g'(t) = 7968t + 368/5

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Complete Question:

A car is traveling at 112 km/h due south at a point 4/5 kilometer north of an intersection_ police, the car Is traveling at 96 km/h due west to at point 2/5 kilometer due cust of the same intersection. At that instant; the radar in the police car measures the rate at which the distance between the two cars [ changing: What does the radar gun register?

The quadratic function in vertex form that passes through the point (1, -3) is: f(x) = (x - 1)²  - 3

What is vertex form?

Vertex form is a way of expressing a quadratic function of the form:

f(x) = a(x - h)² + k

where (h, k) is the vertex of the parabola, and a is a constant that determines the shape and direction of the parabola.

The quadratic function in vertex form is given by:

f(x) = a(x - h)²  + k

where (h, k) is the vertex of the parabola.

We are given the point (1, -3), which lies on the parabola. This means that:

f(1) = -3

Substituting x = 1 into the vertex form of the equation, we get:

f(1) = a(1 - h)²  + k

-3 = a(1 - h)²  + k

Since we don't know the value of h or a, we can't solve for k directly. However, we can use the vertex form of the equation to find the values of h and k.

The vertex of the parabola is the point (h, k). Since the parabola passes through the point (1, -3), we know that the vertex lies on the axis of symmetry, which is the vertical line x = 1.

Therefore, the x-coordinate of the vertex is h = 1. Substituting this into the equation above, we get:

-3 = a(1 - 1)²  + k

-3 = a(0) + k

k = -3

Now that we know the value of k, we can substitute it back into the equation above and solve for a:

-3 = a(1 - h)²  + k

-3 = a(1 - 1)²  + (-3)

-3 = a(0) - 3

a = 1

Therefore, the quadratic function in vertex form that passes through the point (1, -3) is:

f(x) = (x - 1)²  - 3

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