If h is the height, I the slant height and `r_(1), r_(2)` radii of the circular bases of the frustum of a cone, then slant height of the frustum = `sqrt(( r_(1) - r_(2))^(2) + h^(2))`. Height of the cone of which the frustum is a part = (h`r_(1)`/(`r_(1) - r_(2)`).We have a bucket in the form of frustum of a cone in which h = 8 cm, `r_(1)` = 9 cm and `r_(2)` = 3 cm. Find its volume.

Options

A. `321picm^3`
B. `312picm^3`
C. `108picm^3`
D. `324picm^3`

Answer

To find the volume of a bucket in the form of a frustum of a cone, we will use the given dimensions and the formula for the volume of a frustum. Let's go through the solution step by step.

### Step 1: Identify the given values
We are given:
- Height (h) = 8 cm
- Radius of the top base (r₁) = 9 cm
- Radius of the bottom base (r₂) = 3 cm

### Step 2: Write the formula for the volume of a frustum of a cone
The formula for the volume (V) of a frustum of a cone is given by:
\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 r_2) \]

### Step 3: Substitute the values into the formula
Now, substitute the values of h, r₁, and r₂ into the formula:
\[ V = \frac{1}{3} \pi (8) \left( (9)^2 + (3)^2 + (9)(3) \right) \]

### Step 4: Calculate the squares and products
Calculate \( (9)^2 \), \( (3)^2 \), and \( (9)(3) \):
- \( (9)^2 = 81 \)
- \( (3)^2 = 9 \)
- \( (9)(3) = 27 \)

Now substitute these values back into the equation:
\[ V = \frac{1}{3} \pi (8) \left( 81 + 9 + 27 \right) \]

### Step 5: Simplify the expression inside the parentheses
Now, add the values inside the parentheses:
\[ 81 + 9 + 27 = 117 \]
Thus, the equation becomes:
\[ V = \frac{1}{3} \pi (8) (117) \]

### Step 6: Multiply the values
Now, multiply \( \frac{1}{3} \) by \( 8 \) and then by \( 117 \):
\[ V = \frac{8 \times 117}{3} \pi \]

### Step 7: Calculate \( 8 \times 117 \)
Calculate \( 8 \times 117 \):
\[ 8 \times 117 = 936 \]
Now, divide by \( 3 \):
\[ \frac{936}{3} = 312 \]

### Step 8: Write the final volume
Thus, the volume of the frustum of the cone is:
\[ V = 312 \pi \text{ cm}^3 \]

### Final Answer
The volume of the bucket in the form of a frustum of a cone is \( 312 \pi \text{ cm}^3 \).

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