Area.The radius,r,of a circle is decreasing at a rate of 4 centimeters per minute. Find the rate of change of area,A,when the radius is 6.

A)
B)
C)
D)
E)

Question

​Area.The radius,r,of a circle is decreasing at a rate of 4 centimeters per minute. Find the rate of change of area,A,when the radius is 6.
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A) ​ 
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

We know that the area of a circle is given by the formula A=πr². We can find the rate of change of area with respect to time by taking the derivative of both sides of this equation with respect to time:

dA/dt = d/dt (πr²)

Using the chain rule, we can rewrite this as:

dA/dt = 2πr (dr/dt)

Now we can substitute the given values:

r = 6 cm
dr/dt = -4 cm/min (because the radius is decreasing)

dA/dt = 2π(6) (-4) = -48π cm²/min

So the rate of change of area when the radius is 6 cm and decreasing at a rate of 4 cm/min is -48π cm²/min. This corresponds to answer choice D).

​AreaThe Radius,r,of a Circle Is Decreasing at a Rate of 4

AreaThe Radius,r,of a Circle Is Decreasing at a Rate of 4