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The height of a rectangle is increasing at a rate of 11 centimeters per hour and the width of the rectangle is decreasing at a rate of 9 centimeters per hour.
At a certain instant, the height is 3 centimeters and the width is 8 centimeters.
What is the rate of change of the area of the rectangle at that instant (in square centimeters per hour)?
Choose 1 answer:
(A) -61
(B) 61
(C) 99
(D) -99

The height of a rectangle is increasing at a rate of $11$ centimeters per hour and the width of the rectangle is decreasing at a rate of $9$ centimeters per hour. $\newline$ At a certain instant, the height is $3$ centimeters and the width is $8$ centimeters. $\newline$ What is the rate of change of the area of the rectangle at that instant (in square centimeters per hour)? $\newline$ Choose $1$ answer: $\newline$ (A) $-61$ $\newline$ (B) $61$ $\newline$ (C) $99$ $\newline$ (D) $-99$

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Find instantaneous rates of change

Full solution

Q. The height of a rectangle is increasing at a rate of

11

centimeters per hour and the width of the rectangle is decreasing at a rate of

9

centimeters per hour.

\newline

At a certain instant, the height is

3

centimeters and the width is

8

centimeters.

\newline

What is the rate of change of the area of the rectangle at that instant (in square centimeters per hour)?

\newline

Choose

1

answer:

\newline

(A)

-61

\newline

(B)

61

\newline

(C)

99

\newline

(D)

-99

Calculate Initial Area: First, let's find the initial area of the rectangle. The area is height times width. $\newline$ So, Area = Height $\times$ Width = $3\, \text{cm}$ $\times$ $8\, \text{cm}$ = $24\, \text{cm}^2$ .
Calculate Rate of Change: Now, we need to calculate the rate of change of the area. The rate of change of the area is the sum of the product of the height's rate of change and the current width, and the product of the width's rate of change and the current height. $\newline$ So, Rate of change of Area = $(\text{Height's rate of change} \times \text{Current Width}) + (\text{Width's rate of change} \times \text{Current Height})$ .
Calculate Rate of Change: Plugging in the values, we get Rate of change of Area = $11 \text{ cm/hour} \times 8 \text{ cm}$ + $-9 \text{ cm/hour} \times 3 \text{ cm}$ .
Calculate Final Rate: Now, let's do the math: Rate of change of Area = $11 \times 8$ + $-9 \times 3$ = $88$ - $27$ = $61$ $\text{cm}^2/\text{hour}$ .

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