Question

a rectangular prism has side lengths of 4 inches, 6 inches, and 7 inches. if all side lengths are multiplied by 5, what happens to the surface area? a. it remains the same. b. it is multiplied by 5. c. it is multiplied by 25. d. it is multiplied by 125.

Explanation:

Step1: Recall surface - area formula for rectangular prism

The surface - area formula of a rectangular prism with length $l$, width $w$, and height $h$ is $S = 2(lw+lh + wh)$.

Step2: Consider the original and new dimensions

Let the original dimensions be $l_1$, $w_1$, $h_1$ and the new dimensions be $l_2 = 5l_1$, $w_2=5w_1$, $h_2 = 5h_1$.

Step3: Calculate the new surface - area

The new surface - area $S_2=2(l_2w_2 + l_2h_2+w_2h_2)$. Substitute $l_2 = 5l_1$, $w_2 = 5w_1$, $h_2 = 5h_1$ into the formula:
\[

$$\begin{align*} S_2&=2(5l_1\times5w_1+5l_1\times5h_1 + 5w_1\times5h_1)\\ &=2\times25(l_1w_1 + l_1h_1+w_1h_1)\\ &=25\times2(l_1w_1 + l_1h_1+w_1h_1) \end{align*}$$

\]
Since the original surface - area $S_1 = 2(l_1w_1 + l_1h_1+w_1h_1)$, the new surface - area $S_2$ is 25 times the original surface - area.

Answer:

C. It is multiplied by 25.