The radius of a circle is increasing at a rate of 3 centimeters per second. At a certain instant, the radius is 8 centimeters. What is the rate of change of the area of the circle at that instant (in square centimeters per second)? Choose 1 answer: (A) 192 pi (B) 64 pi (C) 9pi (D) 48 pi

Circle Area Formula: First, we need to know the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius.Derivative Calculation: Next, we need to find the derivative of the area with respect to time, dA/dt, since we're looking for the rate of change of the area. This means we'll use the chain rule to differentiate A = πr^2 with respect to time.Chain Rule Application: The derivative of A with respect to r is dA/dr = 2πr. Then, since the radius is changing with respect to time, we multiply by dr/dt, which is the rate of change of the radius with respect to time.Rate of Change Calculation: We know that dr/dt = 3 cm/s (given in the problem). Now we just plug in the values: dA/dt = 2πr * dr/dt.Final Result Calculation: Substitute r = 8 cm and dr/dt = 3 cm/s into the equation: dA/dt = 2π * 8 * 3.Calculate the rate of change of the area: dA/dt = 2 * π * 8 * 3 = 48π square centimeters per second.

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The radius of a circle is increasing at a rate of 3 centimeters per second.
At a certain instant, the radius is 8 centimeters.
What is the rate of change of the area of the circle at that instant (in square centimeters per second)?
Choose 1 answer:
(A)
192 pi
(B)
64 pi
(C)
9pi
(D)
48 pi

The radius of a circle is increasing at a rate of $3$ centimeters per second. $\newline$ At a certain instant, the radius is $8$ centimeters. $\newline$ What is the rate of change of the area of the circle at that instant (in square centimeters per second)? $\newline$ Choose $1$ answer: $\newline$ (A) $192 \pi$ $\newline$ (B) $64 \pi$ $\newline$ (C) $9 \pi$ $\newline$ (D) $48 \pi$

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Math Problems

Grade 6

Area of quadrilaterals and triangles: word problems

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

3

centimeters per second.

\newline

At a certain instant, the radius is

8

centimeters.

\newline

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

\newline

Choose

1

answer:

\newline

(A)

192 \pi

\newline

(B)

64 \pi

\newline

(C)

9 \pi

\newline

(D)

48 \pi

Circle Area Formula: First, we need to know the formula for the area of a circle, which is $A = \pi r^2$ , where $A$ is the area and $r$ is the radius.
Derivative Calculation: Next, we need to find the derivative of the area with respect to time, $\frac{dA}{dt}$ , since we're looking for the rate of change of the area. This means we'll use the chain rule to differentiate $A = \pi r^2$ with respect to time.
Chain Rule Application: The derivative of $A$ with respect to $r$ is $\frac{dA}{dr} = 2\pi r$ . Then, since the radius is changing with respect to time, we multiply by $\frac{dr}{dt}$ , which is the rate of change of the radius with respect to time.
Rate of Change Calculation: We know that $\frac{dr}{dt} = 3\,\text{cm/s}$ (given in the problem). Now we just plug in the values: $\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}$ .
Final Result Calculation: Substitute $r = 8\,\text{cm}$ and $\frac{dr}{dt} = 3\,\text{cm/s}$ into the equation: $\frac{dA}{dt} = 2\pi \times 8 \times 3$ .
Final Result Calculation: Substitute $r = 8$ cm and $\frac{dr}{dt} = 3$ cm/s into the equation: $\frac{dA}{dt} = 2\pi \times 8 \times 3$ .Calculate the rate of change of the area: $\frac{dA}{dt} = 2 \times \pi \times 8 \times 3 = 48\pi$ square centimeters per second.