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Calculus/Differential Calculus

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Expert: - 6/21/2010


Question
A right circular cylinder is inscribed in a sphere of radius 10cm. If the base of the cylinder has a radius of x(cm), express its value as a function of x.

Answer
I'm not sure what "its value" is, for it could be the area or volume.
This said, I will find each separately.

Volumne
If the base has radius x and the sphere has radius 10, it is known that x² + y² = 10²,
or y = √(100-x²).  This is a descrription of half of the height, so the height is h = 2y.
The volume is πx²h, so the volume is πx²2y, which is  V = 2πx²√(100-x²).

Area
The area of the cylinder is the top plus the bottom plus the edge.
The top has area πx², the bottom has the same area, and the outside has area 2πx(2y) = 4πxy.
Putting in y gives 4πx√(100-x²).  Thus, the total area is 2πx² + 4πx√(100-x²).  Now with that,
2πx can be factored out, gives A = 2πx(x + 2√(100-x²)).

Since this is "Differential Calculus," do you want the derivative?  Which one?
V can be differentiated with the product rule of 2πx² and √(100-x²).  The √(100-x²) requires the chain rule and noting that the squareroot is the same as to the 1/2 power.  For the area, take the derivatiee before factoring the 2πx out, giving V'(x) = 4πx - 2x/√(100-x²).

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