Chapter 12.8, Problem 15IP

(a)

To determine

To find: The surface area of the cube.

Answer to Problem 15IP

The surface area of the cube is $150{\text{in}}^2$ .

Explanation of Solution

Given Information:

Consider the given cube.

  

Calculation:

Write the expression to calculate the surface area of the cube.

  $\text{Surface}\text{area}=6a^2$

Where,

• $a$ is the length of the edge.

Substitute the values in the above expression.

  $\begin{array}{c}\text{Surface}\text{area}=6{\left(5\right)}^2\\ =150{\text{in}}^2\end{array}$

Conclusion:

Therefore, the surface area of the cube is $150{\text{in}}^2$ .

(b)

To determine

To complete: The table.

Answer to Problem 15IP

The complete table is given below.

Explanation of Solution

Given Information:

Consider the given table.

Scale factor of Dilation	Original	$\times 2$	$\times 3$
Length of side
Surface area

Calculation:

Write the expression to calculate the surface area of the cube.

  $\text{Surface}\text{area}=6a^2$

Where,

• $a$ is the length of the edge.

From part (a), the original side length is 5 inches and the surface area is 150 square inches. Complete the table.

Scale factor of Dilation	Original	$\times 2$	$\times 3$
Length of side	5	10	15
Surface area	150	600	1350

Conclusion:

Therefore, the complete table is shown above.

(c)

To determine

To compare: The original surface area with the surface area by a scale factor 2 and by a scale factor of 3.

Answer to Problem 15IP

The surface area after a scale factor 2 is 4 times the original surface area and the surface area after a scale factor of 3 is 9 times the original surface area.

Explanation of Solution

Given Information:

Consider the table from part (b).

Scale factor of Dilation	Original	$\times 2$	$\times 3$
Length of side	5	10	15
Surface area	150	600	1350

Calculation:

Write the expression to calculate the surface area of the cube.

  $\text{Surface}\text{area}=6a^2$

Where,

• $a$ is the length of the edge.

From part (b), it is clear that the surface area after a scale factor 2 is 4 times the original surface area and the surface area after a scale factor of 3 is 9 times the original surface area.

Conclusion:

Therefore, the surface area after a scale factor 2 is 4 times the original surface area and the surface area after a scale factor of 3 is 9 times the original surface area.

(d)

To determine

To check: The stated relationship exist with any rectangular prism.

Answer to Problem 15IP

Yes, the relationship exist and it is similar.

Explanation of Solution

Given Information:

From part (c), the surface area after a scale factor 2 is 4 times the original surface area and the surface area after a scale factor of 3 is 9 times the original surface area.

Calculation:

Write the expression to calculate the surface area of the cube.

  $\text{Surface}\text{area}=6a^2$

Where,

• $a$ is the length of the edge.

Let’s consider the length of the rectangular is $l$ , width is $w$ , and height is $h$ . Write an expression to calculate the surface area of the rectangular prism.

  $\text{Surface}\text{area}=2\left(lw+wh+hl\right)$

Calculate the surface area when the sides of the prism is scaled by a factor of 2.

  $\begin{array}{c}\text{Surface}\text{area}=2\left(\left(2l\right)\left(2w\right)+\left(2w\right)\left(2h\right)+\left(2h\right)\left(2l\right)\right)\\ =2\left(4lw+4wh+4hl\right)\\ =4\left[2\left(lw+wh+hl\right)\right]\end{array}$

From the above, it is clear that surface area of the rectangular prism is 4 times the original surface area of the rectangular prism when scaled by a factor of 2. This shows the similar relation.

Conclusion:

Therefore, the relationship exist and it is similar.