The radius and height of a right circular cone are 12 cm and 15 cm respectively. The cone is cut by a plane parallel to its base so as to divide it into two parts. The volume of the frustum (the lower part) of the cone is `320π cm^(3)` Find the radius of the upper circular of the frustum

Options

A. `root (2) (30 ) cm`
B. `3 root (2) (15) cm`
C. `4 root (3)( 15) cm`
D. None of these

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the volume of the original cone
The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

Given:
- Radius \( r = 12 \) cm
- Height \( h = 15 \) cm

Substituting the values:
\[ V = \frac{1}{3} \pi (12)^2 (15) \]
\[ V = \frac{1}{3} \pi (144) (15) \]
\[ V = \frac{1}{3} \pi (2160) \]
\[ V = 720 \pi \, \text{cm}^3 \]

### Step 2: Set up the volume of the frustum
The volume of the frustum \( V_f \) is given as \( 320 \pi \, \text{cm}^3 \).

### Step 3: Calculate the volume of the smaller cone
The volume of the smaller cone \( V_s \) can be calculated using the total volume of the original cone and the volume of the frustum:
\[ V_s = V - V_f \]
Substituting the known values:
\[ V_s = 720 \pi - 320 \pi = 400 \pi \, \text{cm}^3 \]

### Step 4: Relate the dimensions of the smaller cone to the original cone
Let \( r' \) be the radius of the upper circular face of the frustum, and \( h' \) be the height of the smaller cone. Since the cones are similar, we can set up the following ratio:
\[ \frac{r'}{h'} = \frac{12}{15} \]
This simplifies to:
\[ \frac{r'}{h'} = \frac{4}{5} \]
Thus, we can express \( h' \) in terms of \( r' \):
\[ h' = \frac{5}{4} r' \]

### Step 5: Substitute \( h' \) into the volume formula for the smaller cone
The volume of the smaller cone can also be expressed as:
\[ V_s = \frac{1}{3} \pi (r')^2 h' \]
Substituting \( h' \):
\[ V_s = \frac{1}{3} \pi (r')^2 \left(\frac{5}{4} r'\right) \]
\[ V_s = \frac{5}{12} \pi (r')^3 \]

### Step 6: Set the volume of the smaller cone equal to \( 400 \pi \)
Now, we equate the volume of the smaller cone to \( 400 \pi \):
\[ \frac{5}{12} \pi (r')^3 = 400 \pi \]
Dividing both sides by \( \pi \):
\[ \frac{5}{12} (r')^3 = 400 \]
Multiplying both sides by \( \frac{12}{5} \):
\[ (r')^3 = 400 \times \frac{12}{5} \]
\[ (r')^3 = 960 \]

### Step 7: Solve for \( r' \)
Taking the cube root:
\[ r' = \sqrt[3]{960} \]
Calculating the cube root:
\[ r' \approx 10 \, \text{cm} \]

Thus, the radius of the upper circular face of the frustum is approximately \( 10 \, \text{cm} \).