   Chapter 3.7, Problem 31E

Chapter
Section
Textbook Problem

# A right circular cylinder is inscribed in a sphere of radius r . Find the largest possible volume of such a cylinder.

To determine

To find:

The largest possible volume of a cylinder which is inscribed in a sphere

Explanation

1) Concept:

First derivative test:

Suppose c is a critical number of a continuous function f defined on an interval.

(a) If f'(x)>0 for all x<c and f'(x)<0 for all x>c, then f(c) is the absolute maximum value of f.

(b) If f'(x)<0 for all x<c and f'(x)>0 for all x>c, then f(c) is the absolute minimum value of f.

2) Formula:

Volume of right circular cylinder=V=πx2h

Where, x=radius and h=height of right circular cylinder

3) Given:

A right circular cylinder is inscribed in a sphere of radius r

4) Calculation:

Volume of right circular cylinder=V=πx2h

Where, r=radius and h=height of right circular cylinder

r2=h22+x2

x2=r2-h24

Taking square root from both sides,

x=r2-h24 (1)

V=πx2h=V=πr2-h24h=πr2h-πh34

Differentiating the above equation w

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