Mathematics : asked on c6ee6c36ad72467f06e3499e78d0eb11

03.05.2020

A) What is the formula will you use to solve the problem
B) What is value for x
C) What are the dimensions of the base
D) What is the area of the base

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02.12.2022, solved by verified expert

Nitin Arora

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Mathematics, English, Physics. Research Scholar at Indian Institute of Technology, Roorkee

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Answer:

see below

Step-by-step explanation:

A) The formula for the volume of a pyramid is V=Bh where B is the base area and h is the height.

Therefore for the given prism

Equation is 216 = x(x+3)*18

x(x+3) = 12

x = 2.27

C) dimenstions are 2.27 x 5.27

d) Area of the base = x(x+3) =2.27x5.27= 11.96

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Faq

Mathematics

Allyson is taking a class in which the grade is determined by the mean (average) of four tests. She has the following test scores so far in the course: 68, 87, and 90 and asks you to help her figure out what her last test score must be to get an 85 in the class. She tells you that a particular formula is used to figure the final grade of each student in the class. Let the variable x equal the last test score. a) Write the formula needed to solve this problem. b) Substitute the values and the variable into the formula. c) Solve for the test score

Step-by-step answer

P Answered by PhD

Step-by-step explanation:

Test scores:

68, 87, 90 and x

Average:

(68+87+90+x)/4 = 85245 + x = 4*85245 + x = 340x = 340 -245x = 95

Required score is 95

Mathematics

A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with a lid. (a) Write a formula V(x) for the volume of the box. (b) Find the domain of V for the problem situation. (c) Use ANALYTICAL method (my teacher said don t graph) to find the maximum volume and the value of x that gives it. SEE IMAGE!!!

Step-by-step answer

P Answered by PhD

$V(x) = 66.02\ \text{in}^3 \ \vert \ _x_=_1_._9_6$

Step-by-step explanation:

Part (a)

The volume of a rectangular prism is found using the formula: V = lwh.

In order to write a formula in terms of the variable x, we need to write expressions for the length, width, and height of the rectangle.

We can say the length is the bottom of the base of the prism. To find this value, we can subtract x and x from 15 and divide this by 2, since there are two equal rectangles (base and lid).

Length: $\frac{15-2x}{2} =7.5-x$

We can say the width is the side of the base of the prism. To find this value, we can subtract 2x from 10.

Width:

The height of the prism can be x, which is the labeled length of the rectangle next to the base and lid.

Height:

Now we are able to write a formula for volume in terms of x.

Part (b)

The domain of the volume is where x > 0, 2x < 10, and 2x < 15.

This is because our expressions for length ( $\frac{15-2x}{2}$ ), width (10-2x), and height (x) cannot go below 0, because you cannot have a negative value for measurement - realistically.

Therefore, if we take into account all of these restrictions on x, the domain is where 0 < x < 5.

x cannot be > 5 since that would not satisfy 2x < 10.

The domain of V for this problem situation is D: (0, 5).

Part (c)

In order to find the maximum volume of the rectangular prism using our formula we derived, we can use the idea of optimization in calculus.

Start by taking the derivative of V(x).

Set the derivative equal to 0. This gives us the critical point(s) in order to determine the extreme values of the function, aka where the max and min occur.

Solve for x by using the quadratic formula.

$x=\frac{-(-50)\pm\sqrt{(-50)^2-4(6)(75)} }{2(6)}$ $x=\frac{50\pm\sqrt{2500-1800}}{12}$ $x=\frac{50\pm\sqrt{700} }{12}$

Input this into your calculator and you should get:

$x=6.37, \ 1.96$

We are going to use x = 1.96 since 6.37 is NOT in the domain of V(x).

$\boxed{x=1.96}$

Now, since this value of x is going to give the maximum volume of this rectangular prism, or cardboard box, we can plug it back into the V(x) equation for volume to determine the maximum volume of the box.

$V(1.96) = 66.02\ \text{in}^3 \ \vert \ _x_=_1_._9_6$

The maximum volume of the cardboard box is 66.02 in³ and the value of x that gives this is 1.96.

Mathematics

Step-by-step answer

P Answered by PhD

$V(x) = 66.02\ \text{in}^3 \ \vert \ _x_=_1_._9_6$

Step-by-step explanation:

Part (a)

The volume of a rectangular prism is found using the formula: V = lwh.

In order to write a formula in terms of the variable x, we need to write expressions for the length, width, and height of the rectangle.

We can say the length is the bottom of the base of the prism. To find this value, we can subtract x and x from 15 and divide this by 2, since there are two equal rectangles (base and lid).

Length: $\frac{15-2x}{2} =7.5-x$

We can say the width is the side of the base of the prism. To find this value, we can subtract 2x from 10.

Width:

The height of the prism can be x, which is the labeled length of the rectangle next to the base and lid.

Height:

Now we are able to write a formula for volume in terms of x.

Part (b)

The domain of the volume is where x > 0, 2x < 10, and 2x < 15.

This is because our expressions for length ( $\frac{15-2x}{2}$ ), width (10-2x), and height (x) cannot go below 0, because you cannot have a negative value for measurement - realistically.

Therefore, if we take into account all of these restrictions on x, the domain is where 0 < x < 5.

x cannot be > 5 since that would not satisfy 2x < 10.

The domain of V for this problem situation is D: (0, 5).

Part (c)

In order to find the maximum volume of the rectangular prism using our formula we derived, we can use the idea of optimization in calculus.

Start by taking the derivative of V(x).

Set the derivative equal to 0. This gives us the critical point(s) in order to determine the extreme values of the function, aka where the max and min occur.

Solve for x by using the quadratic formula.

$x=\frac{-(-50)\pm\sqrt{(-50)^2-4(6)(75)} }{2(6)}$ $x=\frac{50\pm\sqrt{2500-1800}}{12}$ $x=\frac{50\pm\sqrt{700} }{12}$

Input this into your calculator and you should get:

$x=6.37, \ 1.96$

We are going to use x = 1.96 since 6.37 is NOT in the domain of V(x).

$\boxed{x=1.96}$

$V(1.96) = 66.02\ \text{in}^3 \ \vert \ _x_=_1_._9_6$

The maximum volume of the cardboard box is 66.02 in³ and the value of x that gives this is 1.96.

Mathematics

The last time Robert filled up his car with gas, he paid $24.50 for 7 gallons. This time, he needs 15 gallons. If the price is the same, how much will he pay?

Step-by-step answer

P Answered by PhD