The radius of a circle is decreasing at a rate of 6.5 meters per minute. At a certain instant, the radius is 12 meters. What is the rate of change of the area of the circle at that instant (in square meters per minute)? Choose 1 answer: (A) -156 pi (B) -144 pi (C) -288 pi (D) -42.25 pi

Circle Area Formula: The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.Rate of Change Derivation: To find the rate of change of the area, we need to differentiate the area with respect to time (t). So we get dA/dt = 2πr * dr/dt.Radius Rate of Change: We know the radius is decreasing at a rate of 6.5 meters per minute, so dr/dt = -6.5 meters/minute.Instant Calculation: At the instant when the radius is 12 meters, we plug r = 12 meters and dr/dt = -6.5 meters/minute into the formula dA/dt = 2πr * dr/dt.Final Result: So, dA/dt = 2π * 12 meters * (-6.5 meters/minute) = -156π square meters per minute.

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The radius of a circle is decreasing at a rate of 6.5 meters per minute.
At a certain instant, the radius is 12 meters.
What is the rate of change of the area of the circle at that instant (in square meters per minute)?
Choose 1 answer:
(A)
-156 pi
(B)
-144 pi
(C)
-288 pi
(D)
-42.25 pi

The radius of a circle is decreasing at a rate of $6$ . $5$ meters per minute. $\newline$ At a certain instant, the radius is $12$ meters. $\newline$ What is the rate of change of the area of the circle at that instant (in square meters per minute)? $\newline$ Choose $1$ answer: $\newline$ (A) $-156 \pi$ $\newline$ (B) $-144 \pi$ $\newline$ (C) $-288 \pi$ $\newline$ (D) $-42.25 \pi$

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Math Problems

Grade 6

Area of quadrilaterals and triangles: word problems

Full solution

Q. The radius of a circle is decreasing at a rate of

6

5

meters per minute.

\newline

At a certain instant, the radius is

12

meters.

\newline

What is the rate of change of the area of the circle at that instant (in square meters per minute)?

\newline

Choose

1

answer:

\newline

(A)

-156 \pi

\newline

(B)

-144 \pi

\newline

(C)

-288 \pi

\newline

(D)

-42.25 \pi

Circle Area Formula: The formula for the area of a circle is $A = \pi r^2$ , where $A$ is the area and $r$ is the radius.
Rate of Change Derivation: To find the rate of change of the area, we need to differentiate the area with respect to time $t$ . So we get $\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$ .
Radius Rate of Change: We know the radius is decreasing at a rate of $6.5$ meters per minute, so $\frac{dr}{dt} = -6.5$ meters/minute.
Instant Calculation: At the instant when the radius is $12$ meters, we plug $r = 12$ meters and $\frac{dr}{dt} = -6.5$ meters/minute into the formula $\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$ .
Final Result: So, $\frac{dA}{dt} = 2\pi \times 12 \text{ meters} \times (-6.5 \text{ meters/minute}) = -156\pi \text{ square meters per minute}.$