The spherical balloon is inflated at the rate of 10 m³/sec. find the rate at which the surface area is increasing when the radius of the sphere is 3m?

The balloon has a volume dependent on its radius :

Differentiating with respect to time gives

If the volume is increasing at a rate of 10 cubic m/s, then at the moment the radius is 3 m, it is increasing at a rate of

The surface area of the balloon is

and differentiating gives

so that at the moment the radius is 3 m, its area is increasing at a rate of

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