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If the radius of the circle changes at the rate of , at what rate does the circle's area change when the radius is 10 m?

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  1. 40 m2/sec
  2. 30 m2/sec
  3. -30 m2/sec
  4. -40 m2/sec

Answer (Detailed Solution Below)

Option 4 : -40 m2/sec
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Detailed Solution

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Concept:

  • The area of a circle with r units, is: πr2 sq. units.
  • The rate of change of the value of a function f(x) with respect to a variable t, is given by: .

  • Chain Rule of Derivatives: .

  • .

 

Calculation:

Let's say that the radius of the circle is r meters and let t be the time in seconds.

It is given that .

Now, Area of the circle: A = πr2 m2.

Using the chain rule of derivatives:

⇒ 

⇒ 

When the radius is 10 m, the rate of change of area will be: -40 m2/sec.

NOTE: A negative value of the rate of change indicates a decrease.

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