A cone of volume 54π is cut by a plane parallel to the base, 1/3 of the way up the height of the cone (from the base). Find the volume of the resulting frustum.

Correct answer: Vf = 38 π

Step-by-step explanation:

Given:

Vc = 54 π

The ratio of the height of the cone and the height of the frustum is:

Hc : Hf = 3 : 1 ⇒ Hc = 3 · Hf

The ratio of the radius R of the base of the cone to the radius r of the upper base of the frustum is:

R : r = Hc : (Hc - Hf) ⇒ R : r = (3· Hf) : (3 · Hf - Hf) ⇒ R : r = 3 · Hf : 2 · Hf

R : r = 3 : 2 ⇒ r = (2/3) · R

The formula for calculating the volume of a cone is:

Vc = (R² · π · Hc) / 3 = (R² · π · 3 · Hf) / 3 = R² · π · Hf

The formula for calculating the volume of a frustum is:

Vf = (π · Hf · (R² + R · r + r²)) / 3 = (π · Hf · (R² + R · (2/3) · R + ((2/3) R) ²)) / 3

Vf = (π · Hf · (R² + (2/3) ·R² + (4/9) · R²)) / 3 ⇒

Vf = (π · Hf · ((9/9) R² + (6/9) ·R² + (4/9) · R²)) / 3 ⇒

Vf = (π · Hf · (19/9) · R²) / 3 = (19/27) · R² · π · Hf

Vc / Vf = (R² · π · Hf) / ((19/27) · R² · π · Hf)

R², π and Hf were shortened and we get:

Vc / Vf = 27 / 19 ⇒ Vf = (19 · Vc) / 27 ⇒

Vf = 19 · 54 · π / 27 = 38 π

Vf = 38 π

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