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## Example Questions

### Example Question #1 : How To Find The Length Of An Edge Of A Prism

For a box to fit inside the cupboard, the sum of the height and the perimeter of the box must, at most, be 360 cm. If Jenn has a box that has a height of 40 cm and a length of 23 cm, what is the greatest possible width of the box?

**Possible Answers:**

137 cm

13 cm

0.4 cm

207 cm

297 cm

**Correct answer:**

137 cm

First we write out the equation we are given. *H* + (2*L* +2*W*) = 360. *H *= 40 and *L *= 23

40 + (2(23) + 2*W*) = 360

40 + (46 + 2*W*) = 360

46 + 2*W* = 320

2*W* = 274

*W* = 137

### Example Question #2 : Prisms

The volume of a rectangular prism is 80 cm^{3}. The length, width, and height of the prism are each an integer number of cm. If the dimensions form three terms of an arithmetic sequence, find the average of the three dimensions.

**Possible Answers:**

6

4

8

7

5

**Correct answer:**

5

Method 1:

Trial and error to find a combination of factors of 80 that differ by the same amount will eventually yield 2, 5, 8. The average is 5.

Method 2:

Three terms of an arithmetic sequence can be written as x, x+d, and x+2d. Multiply these together using the distributive property to find the volume and the following equation results:

x^{3} + 3dx^{2} + 2d^{2}x - 80 = 0

Find an integer value of x that creates an integer solution for d. Try x=1 and we see the equation 1 + 3d + 2d^{2} - 80 = 0 or 2d^{2} + 3d -79 = 0. The determinant of this quadratic is 641, which is not a perfect square. Therefore, d is not an integer when x=1.

Try x=2 and we see the equation 8 + 12d + 4d^{2} - 80 = 0 or d^{2} + 3d - 18 = 0. This is easily factored to (d+6)(d-3)=0 so d=-6 or d=3. Since a negative value of d will result in negative dimensions of the prism, d must equal 3. Therefore, when substituting x=2 and d=3, the dimensions x, x+d, and x+2d become 2, 5, and 8. The average is 5.

### Example Question #3 : How To Find The Length Of An Edge Of A Prism

A right rectangular prism has a volume of 64 cubic units. Its dimensions are such that the second dimension is twice the length of the first, and the third is one-fourth the dimension of the second. What are its exact dimensions?

**Possible Answers:**

2 x 4 x 8

1 x 4 x 16

3 x 6 x 12

1 x 2 x 32

4 x 4 x 4

**Correct answer:**

2 x 4 x 8

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 2 * Dim1 = 2x

Dim3: (1/4) * Dim2 = (1/4) * 2x = (1/2) * x

More directly stated, therefore, our dimensions are: x, 2x, and 0.5x. Therefore, the volume is x * 2x * 0.5x = 64, which simplifies to x^{3} = 64. Solving for x, we find x = 4. Therefore, our dimensions are:

x = 4

2x = 8

0.5x = 2

Or: 2 x 4 x 8

### Example Question #3 : Prisms

A right rectangular prism has a volume of 120 cubic units. Its dimensions are such that the second dimension is three times the length of the first, and the third dimension is five times the dimension of the first. What are its exact dimensions?

**Possible Answers:**

2 x 6 x 10

4 x 12 x 20

2 x 5 x 12

None of the other answers

1 x 5 x 24

**Correct answer:**

2 x 6 x 10

Based on our prompt, we can say that the prism has dimensions that can be represented as:

Dim1: x

Dim2: 3 * Dim1 = 3x

Dim3: 5 * Dim1 = 5x

More directly stated, therefore, our dimensions are: x, 3x, and 5x. Therefore, the volume is x * 3x * 5x = 120, which simplifies to 15x^{3} = 120 or x^{3} = 8. Solving for x, we find x = 2. Therefore, our dimensions are:

x = 2

3x = 6

5x = 10

Or: 2 x 6 x 10

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