Answer to Question #147885 in Geometry for anlp

Question

The volume of a frustum of a right circular cone is 52π cm³ . its altitude is 3cm and the measure of its lower radius is three times the measure of its upper radius.

1. What is the radius of the upper base?
2. what is the slant height of the frustum?
3. find the lateral area of the frustum.

Answer 1

V = 52$\pi$ cm³ - The volume of a frustum of a right circular cone;

h = 3cm - the altitude of a frustum of a right circular cone;

R - lower radius of a frustum of a right circular cone;

r - upper radius of a frustum of a right circular cone;

R = 3 * r;

we fill the frustum of a right circular cone to right circular cone so

H - the altitude of right circular cone;

If we use similarity rule that is to say SAS postulate

 SAS Postulate: Two triangles are similar if two of their corresponding sides are in proportion and the angle in between is congruent.

H / (H - h) = R / r

H / (H - h) = 3 * r / r

H = 3 * H - 3 * h

3 * h = 2 * H

H = 1.5 * h -> 1.5 * 3cm = 4.5cm;

V = 1 / 3 * $\pi$ * R² * H - 1 / 3 * $\pi$ * r² * (H - h) = 1 / 3 * $\pi$ * (4.5 * 9 * r² - 1.5 * r²) = 1 / 3 * $\pi$ * 39 * r²;

1 / 3 * $\pi$ * 39 * r² = 52$\pi$ cm³

r = 2cm; R = 3 * r = 6cm

1. What is the radius of the upper base?

the radius of the upper base is r = 2cm;

2.what is the slant height of the frustum?

l - the slant height

l² = h² + (R - r)² = (9 + 16)cm²

l = 5cm

3. find the lateral area of the frustum?

Lateral area of the frustum cone

S = $\pi$ * (R + r) * l = $\pi$ * 8 * 5 = 40cm²