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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
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

Given:

The height of the bigger cone is H.

The height of the smaller cone is h.

Theorem used:

AAA Theorem:

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.

Calculation:

Let the radius of the bigger cone is R and the radius of the smaller cone is r.

From Figure 1, consider ΔEBD and ΔEAC,

BED=AEC(Common angle)EBD=EAC=90°EDB=ECA(Corresponding angle)

ΔEBD and ΔEAC are similar triangle.

By AAA Theorem,

HR=HhrHr=HRhRh=HHrR=HRHrR=HR(Rr)

The volume of the inner cone is V=13πr2h.

Substitute the value of h=HR(Rr) in V,

V=13πr2h=13πr2×HR(Rr)=πH3R(Rr2r2)

Differentiate V with respect to r,

dVdr=πH3R(2Rr3r2)

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