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

since you are given the ratio of the slant height to the radius 5:3 the altitude will have a ratio of 5;4 The radius , altitude and slant form a right triangle

Volume of the cone = (1/3) pir^2h= (1/3)pi (3x)(4x)^2

768Pi= (1/3) pi(12x)^2 = (1/3)pi(144x^2)

x^2 =16 x=4 which makes the radius 12 and slant height 20

lateral area is (pi)(12)(16)= 240pi

Answer 2

V n

nnn

n n n

n n n

n n n

n n n

n n n n n

A C B

This is the cross section of the right circular cone.

Note that the height of the cone VC is perpendicular to the radius of base CB or AC.

Triangle VCB is the right triangle and VB:CB = 5:3.

Therefore, VB:CB:VC = 5:3:4 -> Pythagorean triples

Let r = 3x, h = 4x and s = 5x (s - slanted height)

V = hπr^2/3

V - volume of the cone

h - height of the cone

r - radius of the base

768π = hπr^2/3

768 = hr^2/3

768 = 12x^3

x = 4

h = 4x = 16

r = 3x = 12

s = 5x = 20

Now we need to calculate the area of the exterior surface of the cone.

Imagine, we make a cut from the vertex along the slant height.

We carve out the area of the base circle.

Then we unfold and flatten what's left on the horizontal plane.

We will get a triangle with the base equal to curcumference of the base circle (2πr)

and the height h = 16

Let's calculate the area of this triangle:

Area = (base•height)/2 = (2πr•16)/2 = 3.14•12•16 = 602.88