Urpi was given this problem: $\newline$ The side $b(t)$ of the base of a square prism is decreasing at a rate of $7$ kilometers per minute and the height $h(t)$ of the prism is increasing at a rate of $10$ kilometers per minute. At a certain instant $t_{0}$ , the base's side is $4$ kilometers and the height is $9$ kilometers. What is the rate of change of the surface area $S(t)$ of the prism at that instant? $\newline$ Which equation should Urpi use to solve the problem? $\newline$ Choose $1$ answer: $\newline$ (A) $S(t)=6[b(t)]^{2}$ $\newline$ (B) $S(t)=[b(t)]^{3}$ $\newline$ (C) $S(t)=2[b(t)]^{2}+4 b(t) \cdot h(t)$ $\newline$ (D) $S(t)=[b(t)]^{2} \cdot h(t)$

Full solution

Q. Urpi was given this problem:

\newline

The side

b(t)

of the base of a square prism is decreasing at a rate of

7

kilometers per minute and the height

h(t)

of the prism is increasing at a rate of

10

kilometers per minute. At a certain instant

t_{0}

, the base's side is

4

kilometers and the height is

9

kilometers. What is the rate of change of the surface area

S(t)

of the prism at that instant?

\newline

Which equation should Urpi use to solve the problem?

\newline

Choose

1

answer:

\newline

(A)

S(t)=6[b(t)]^{2}

\newline

(B)

S(t)=[b(t)]^{3}

\newline

(C)

S(t)=2[b(t)]^{2}+4 b(t) \cdot h(t)

\newline

(D)

S(t)=[b(t)]^{2} \cdot h(t)

Understand Problem and Formula: Understand the problem and determine the formula for the surface area of a square prism. A square prism's surface area is composed of the areas of the three pairs of parallel faces: the two square bases and the four rectangular sides. The formula for the surface area of a square prism is $S(t) = 2[b(t)]^2 + 4b(t)h(t)$ , where $b(t)$ is the length of the side of the base and $h(t)$ is the height of the prism.
Choose Correct Equation: Choose the correct equation that represents the surface area of a square prism. $\newline$ From the options given, the correct formula that represents the surface area of a square prism is: $\newline$ (C) $S(t) = 2[b(t)]^2 + 4b(t)h(t)$ $\newline$ This is because the surface area of a square prism is the sum of the area of the two square bases ( $2[b(t)]^2$ ) and the area of the four rectangular sides ( $4b(t)h(t)$ ).