The base radius and height of a cone is 5cm and 25 cm respectively. If the cone is cut parallel to its base at a height of h from the base. If the volume of this frustum is 110 cm3 . Find the radius of smaller cone?

Options

A. `(104)^(1//3)cm`
B. `(104)^(1//2)cm`
C. 5cm
D. None of these

Answer

To find the radius of the smaller cone when a cone is cut parallel to its base, we can follow these steps:

### Step 1: Understand the dimensions of the cone
The original cone has:
- Base radius (R) = 5 cm
- Height (H) = 25 cm

### Step 2: Set up the relationship using similar triangles
When the cone is cut parallel to its base at a height \( h \) from the base, we can denote:
- The radius of the smaller cone (r) = ?
- The height of the smaller cone = \( H - h = 25 - h \)

By the property of similar triangles, we have:
\[ \frac{r}{R} = \frac{H - h}{H} \]
Substituting the known values:
\[ \frac{r}{5} = \frac{25 - h}{25} \]
From this, we can express \( r \) in terms of \( h \):
\[ r = 5 \cdot \frac{25 - h}{25} = \frac{5(25 - h)}{25} = \frac{5(25 - h)}{25} = 5 - \frac{h}{5} \]

### Step 3: Calculate the volume of the frustum
The volume \( V \) of the frustum can be calculated using the formula:
\[ V = \frac{1}{3} \pi h (R^2 + Rr + r^2) \]
Where:
- \( R = 5 \) cm (radius of the larger base)
- \( r = 5 - \frac{h}{5} \) (radius of the smaller base)
- \( h = 25 - h \) (height of the frustum)

Given that the volume of the frustum is 110 cm³, we can substitute the values into the volume formula:
\[ 110 = \frac{1}{3} \pi (25 - h) \left(5^2 + 5(5 - \frac{h}{5}) + (5 - \frac{h}{5})^2\right) \]

### Step 4: Simplify the equation
Calculating the terms:
- \( 5^2 = 25 \)
- \( 5(5 - \frac{h}{5}) = 25 - h \)
- \( (5 - \frac{h}{5})^2 = 25 - 2h + \frac{h^2}{25} \)

Now, substituting these back into the volume equation:
\[ 110 = \frac{1}{3} \pi (25 - h) \left(25 + (25 - h) + (25 - 2h + \frac{h^2}{25})\right) \]
Combine the terms:
\[ 110 = \frac{1}{3} \pi (25 - h) \left(75 - 3h + \frac{h^2}{25}\right) \]

### Step 5: Solve for h
This equation can be solved for \( h \) using algebraic methods or numerical methods. Once \( h \) is found, substitute back to find \( r \).

### Step 6: Find the radius of the smaller cone
Once \( h \) is determined, substitute it back into the equation for \( r \):
\[ r = 5 - \frac{h}{5} \]

### Final Result
The radius of the smaller cone can be calculated based on the value of \( h \).