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The side length of a square is increasing at a rate of 15 millimeters per second.
At a certain instant, the side length is 22 millimeters.
What is the rate of change of the area of the square at that instant (in square millimeters per second)?
Choose 1 answer:
(A) 225
(B) 660
(C) 484
(D) 30

The side length of a square is increasing at a rate of $15$ millimeters per second. $\newline$ At a certain instant, the side length is $22$ millimeters. $\newline$ What is the rate of change of the area of the square at that instant (in square millimeters per second)? $\newline$ Choose $1$ answer: $\newline$ (A) $225$ $\newline$ (B) $660$ $\newline$ (C) $484$ $\newline$ (D) $30$

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Calculate unit rates with fractions

Full solution

Q. The side length of a square is increasing at a rate of

15

millimeters per second.

\newline

At a certain instant, the side length is

22

millimeters.

\newline

What is the rate of change of the area of the square at that instant (in square millimeters per second)?

\newline

Choose

1

answer:

\newline

(A)

225

\newline

(B)

660

\newline

(C)

484

\newline

(D)

30

Area Formula: The area of a square is given by the formula $\text{Area} = \text{side length} \times \text{side length}$ . Let's denote the side length as ' $s$ '. So, $\text{Area} = s^2$ .
Differentiation with Respect to Time: To find the rate of change of the area, we need to differentiate the area with respect to time $t$ . So we get $\frac{d(\text{Area})}{dt} = 2 \times s \times \frac{ds}{dt}$ .
Given Values: We know that $\frac{ds}{dt}$ (the rate of change of the side length) is $15\,\text{mm/s}$ and at the instant we are considering, $s = 22\,\text{mm}$ .
Calculation: Now we plug in the values: $\frac{d(\text{Area})}{dt} = 2 \times 22 \, \text{mm} \times 15 \, \text{mm/s} = 2 \times 330 \, \text{mm}^2/\text{s} = 660 \, \text{mm}^2/\text{s}.$
Final Answer: So, the rate of change of the area of the square at that instant is $660$ square millimeters per second. The correct answer is (B) $660$ .

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