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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h = 1 3 H .

To determine

To show:

The inner cone has maximum volume when h=13H.

Explanation

1) Concept:

i) A critical number of a function f   is a number c in the domain of f  Such that either  f'c=0 or f'c does not exist.

ii) First derivative test for absolute extreme values- suppose that c is a critical number of a continuous function f defined on an interval.

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

If f'x<0 for all x<c and f'x>0 for all >c, then fc is the absolute Minimum value of f.

2) Formula:

Volume of a cone:V=π3r2h

3) Given:

A cone with height h is inscribed in a large cone of height H so that its vertex is at the center of the base of the largest cone.

4) Calculation:

From above figure in  ABC and AEF, A  is common and E=B=90°

Therefore, by AAA test,  ABC and AEF are similar.

By properties of similarity of triangles,

HR=H-hr,

Simplifying above equation for h,

HrR=H-h

h=H-HrR=HR-HrR

h=HRR-r

As volume of inscribed cone is,

Vr=13πr2h=13πr2HRR-r

Vr=πH3RRr2-r3

Now for finding maximum volume, first find critical points.

Therefore, differentiating V with respect to r,

V'r= πH3R2Rr-3r2

V'r= πHr3R2R-3r

So for V'r=0 i

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