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Calculus/Calculus: Related Rates

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Expert: - 4/27/2011


Question
A spherical iron ball is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 8 mL/min, how fast is the outer surface area of the ice decreasing when the outer diameter (ball plus ice) is 20 cm?

Answer
The equation for surface area of a sphere is 4*pi*r^2.
The change is found by the derivative times 8 mL/min.
The derivative is 8*pi*r.
Since the diameter is 20 cm, the radius is 1/2 of this, and that is 10 cm.

Thus, the change is 8*pi*(10 cm)*8 mL/min.
This has an 8, 10, and 8, multiplied together.
Since 8*8 is 64, putting the 10 in there gives 640.
This gives 640*pi cm*mL/min.

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