Question

A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

• A. 232
• B. 256
• C. 264
• D. 243

Answer 1

The Correct Option is D

To find the volume of the original cone, we begin by noting that when a cone is cut by a plane parallel to its base, the two resulting pieces are similar to each other and to the original cone.

The original cone's height is \( H = 27 \) cm. Let's denote the radius of the base of the original cone as \( R \). The smaller cone, which is cut off, has a height of \( 18 \) cm and thus a base radius of \( r \), where the heights and radii of similar cones are proportional: \(\frac{r}{R} = \frac{18}{27} = \frac{2}{3}\). Therefore, \( r = \frac{2}{3}R \).

Next, we calculate the volumes. The volume of the original cone \( V \) is:

\[ V = \frac{1}{3} \pi R^2 H = \frac{1}{3} \pi R^2 \times 27 = 9 \pi R^2 \]

The volume of the smaller cone \( v \) is:

\[ v = \frac{1}{3} \pi r^2 \times 18 = \frac{1}{3} \pi \left(\frac{2}{3}R\right)^2 \times 18 = \frac{1}{3} \pi \times \frac{4}{9} R^2 \times 18 = \frac{24}{27} \pi R^2 = \frac{8}{9} \pi R^2 \]

The volume of the frustum obtained after the cut is the difference between the original cone and the smaller cone:

\[ V - v = 9 \pi R^2 - \frac{8}{9} \pi R^2 = \left(9 - \frac{8}{9}\right) \pi R^2 = \frac{81}{9} \pi R^2 - \frac{8}{9} \pi R^2 = \frac{73}{9} \pi R^2 \]

We know from the problem statement that \( V - v = 225 \) cc:

\[ \frac{73}{9} \pi R^2 = 225 \]

Solve for \( \pi R^2 \):

\[ \pi R^2 = \frac{225 \times 9}{73} = \frac{2025}{73} \]

Therefore, the volume of the original cone is:

\[ V = 9 \pi R^2 = 9 \times \frac{2025}{73} = \frac{2025 \times 9}{73} = \frac{18225}{73} \approx 243 \text{ cc}\]

Thus, the volume of the original cone is \( \boxed{243} \) cc.

Answer 2

 Let the radius of the base of the large cone be: \(3r\).

The height of the upper smaller cone is: \(9\). By symmetry, the radius of the upper cone becomes: \(r\).

 Volume Calculations

• Frustum's Volume (difference of full cone and smaller cone): 

\[V_{\text{frustum}} = \frac{\pi}{3}(9r^2 \cdot 27 - r^2 \cdot 9)\]

• Volume of upper cone: 

\[V_{\text{upper}} = \frac{\pi}{3} \cdot r^2 \cdot 9\]

Given Condition

The difference in their volumes is \(225\) cc:

\[\frac{\pi}{3} \cdot 9 \cdot r^2 \cdot 25 = 225 \Rightarrow \pi \cdot 75r^2 = 225 \Rightarrow r^2 = \frac{3}{\pi}\]

Find Volume of the Large Cone

\[V_{\text{large}} = \frac{\pi}{3} \cdot (9r^2) \cdot 27 = \frac{\pi}{3} \cdot 9 \cdot \frac{3}{\pi} \cdot 27 = 243\]

 Final Answer: 243 cc