Item 15 The value of the surface area (in square centimeters) of the cone is equal to the value of the volume (in cubic centimeters) of the cone. The formula for the surface area S of a right cone is S=πr2+πrl, S=πr2+πrl, where r is the radius of the base and l is the slant height. Find the height of the cone.

The volume of a cone is equal to

V = 1/3πr²h

Since volume and surface area is equal, we have to equate this to the given equation:

πr² + πrl = 1/3πr²h

The relationship between l, r and h is

l = √ (h² + r²)

Substituting and simplifying the equation:

πr√ (h² + r²) = 1/3πr²h - πr²

[√ (h² + r²) = 1/3*rh - r]²

h² + r² = 1/9*r²h² - 2/3*r²h + r²

h² + 2/3*r²h - 1/9*r²h² = 0

h (h + 2/3*r² - 1/9*r²h) = 0

h + 2/3*r² - 1/9*r²h = 0

h - 1/9*r²h = - 2/3*r²

h (1 - 1/9*r²) = - 2/3*r

h = - 2r²/3 (1 - 1/9*r²)