For a right circular cone, the ratio of the slant height to the length of the radius is 5 : 3.

Question

For a right circular cone, the ratio of the slant height to the length of the radius is 5 : 3. If the volume of the cone is 768𝜋 in³, find the lateral area (in square inches) of the cone.

Answer 1

Then the radius is 3x and the slant height (hypotenuse) is 5x

the height of the cone, by pythagorean is 4x...

R=3x, H=4x, slant hypotenuse = 5x...

It is left for you to verify the pythagorean

V = 768 pi = (1/3)pi * R^2 * H

= (1/3) pi * (3x)^2 (4x)

= (1/3) pi * (9x^2)(4x)

= 12 pi x^3

pi cancels out

768 = 12 x^3

64 = x^3

x = 4

so the radius is 12, the height is 16, and the slant is 20

note that (1/3) pi (12^2)(16) = (1/3)(12)(12)(16) pi

= 48*16 pi

= (50-2)16 pi

= (800 - 32) pi

= 768 pi

lateral area is pi * r * sqrt [ h^2 + r^2)

= pi * 12 * sqrt [ 16^2 + 12^2]

= pi * 12 * sqrt ( 256 + 144]

= pi * 12 * sqrt( 400)

= pi * 12 * 20

= 240 pi