Calculas H/w Help

Question

A cylinder is inscribed in a right circular cone of height 7.75 inches and radius (at the base) equal to 5.25 inches. What are the dimensions of such a cylinder which has maximum volume?

Answer 1

Draw and label a diagram!

h = -7.5r / 5.25

V = πr²h

V = πr²( -7.5r / 5.25)

V = -7.5πr³ / 5.25

Determine first derivative, set equal to zero, and solve for r.

Determine volume.

Answer 2

Let

H = 7.75" be the height of the cone,

R = 5.25" be its radius,

h be the height of an inscribed cylinder,

r be the radius of an inscribed cylinder.

The upper disc of the cylinder is the base of a smaller cone with height H-h.

That smaller cone is similar to the given cone, and its features are

reduced by the factor (H-h)/H.

Therefore the radius of the cylinder

r = R(H-h)/H

If give some height h, then the volume of the cylinder

V(h) = hπr² = hπR²(H-h)²/H²

The function V(h) is maximized at a critical point, or at boundary of domain.

The boundaries of the domain are at h = 0 and h = H;

for both V(h) = 0.

Therefore the maximum must occur at a critical point, that is, where

V'(h) = 0 (1)

Calculate

V'(h) = (πR²/H²)((H-h)² - 2h(H-h)) = ((H-h)πR²/H²)(H-h - 2h)

Equation (1) has two solutions:

h = H (which is not a maximum)

h = H/3 (which is the only maximum)

Substituting actual numbers

h = 7.75/3 = 2.583"

r = 5.25×(2/3) = 3.5"

Please note that we did not use the fact that the cone is right circular;

the result holds for any angle at the vertex.