When the area of a circle is increasing twice as fast as the radius, the radius is given by

  r = 1/π

Step-by-step explanation:

The area of a circle is given by ...

  A = πr²

The rate of change of area is ...

  A' = 2πr·r'

We want that rate of change to be twice as fast as the rate of change of radius:

  2r' = 2πr·r'

  1 = πr . . . . . . divide by 2r'

  1/π = r

When the area is changing twice as fast as the radius, the radius is 1/π.

Additional comment

The area and the radius have different units, so their rates of change must be compared numerically. The numerical value will depend on the units used. The radius for a circle whose area is measured in m² will be different than for one whose area is measured in ft².

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