Assume that the radius of a sphere is expanding at a rate of . The volume of a sphere is .
Determine the rate at which the volume is changing with respect to time when .
The volume is changing at a rate of .
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To determine the rate at which the volume of the sphere is changing with respect to time, we will use the formula for the volume of the sphere and apply the chain rule.
Given the volume formula:
we differentiate with respect to :
Calculating :
Now, substituting in our known values: in and in/min.
Now substituting into the equation for :
Numerically calculating to provide a specific answer:
So, the volume is changing at a rate of approximately