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) 48 pi (B) 9pi (C) 64 pi (D) 192 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.Differentiate Area 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 dA/dt = d/dt (pi * r^2).Chain Rule Application: Using the chain rule, dA/dt = 2 * pi * r * dr/dt, where dr/dt is the rate of change of the radius.Rate of Radius Change: We know the radius is increasing at a rate of 3 centimeters per second, so dr/dt = 3 cm/s.Plug in Values: Now we plug in the values: dA/dt = 2 * pi * 8 cm * 3 cm/s.Simplify Calculation: Simplify the calculation: dA/dt = 2 * pi * 24 cm^2/s.Final Result: Finally, we get dA/dt = 48 pi cm^2/s.

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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)
48 pi
(B)
9pi
(C)
64 pi
(D)
192 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) $48 \pi$ $\newline$ (B) $9 \pi$ $\newline$ (C) $64 \pi$ $\newline$ (D) $192 \pi$

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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)

48 \pi

\newline

(B)

9 \pi

\newline

(C)

64 \pi

\newline

(D)

192 \pi

Circle Area Formula: The formula for the area of a circle is $A = \pi \cdot r^2$ , where $A$ is the area and $r$ is the radius.
Differentiate Area 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 $\frac{dA}{dt} = \frac{d}{dt} (\pi \cdot r^2)$ .
Chain Rule Application: Using the chain rule, $\frac{dA}{dt} = 2 \cdot \pi \cdot r \cdot \frac{dr}{dt}$ , where $\frac{dr}{dt}$ is the rate of change of the radius.
Rate of Radius Change: We know the radius is increasing at a rate of $3$ centimeters per second, so $\frac{dr}{dt} = 3 \, \text{cm/s}$ .
Plug in Values: Now we plug in the values: $\frac{dA}{dt} = 2 \cdot \pi \cdot 8 \, \text{cm} \cdot 3 \, \text{cm/s}.$
Simplify Calculation: Simplify the calculation: $\frac{dA}{dt} = 2 \times \pi \times 24 \, \text{cm}^2/\text{s}$ .
Final Result: Finally, we get $\frac{dA}{dt} = 48 \pi \, \text{cm}^2/\text{s}$ .