The slant height and the diameter of the base of a right circular cone are 34 cm and 32 cm . respectively . Its volume (in ` cm^(3)` ) is

Options

A. `3200 pi`
B. `3840 pi`
C. `2560 pi`
D. `2880pi`

Answer

To find the volume of a right circular cone given the slant height (L) and the diameter of the base, we can follow these steps:

### Step 1: Identify the given values
- Slant height (L) = 34 cm
- Diameter of the base = 32 cm

### Step 2: Calculate the radius of the base
The radius (R) is half of the diameter.
\[ R = \frac{\text{Diameter}}{2} = \frac{32 \text{ cm}}{2} = 16 \text{ cm} \]

### Step 3: Use the Pythagorean theorem to find the height (H)
In a right circular cone, the relationship between the slant height (L), height (H), and radius (R) is given by:
\[ L^2 = H^2 + R^2 \]
Substituting the known values:
\[ 34^2 = H^2 + 16^2 \]
Calculating the squares:
\[ 1156 = H^2 + 256 \]
Now, isolate \(H^2\):
\[ H^2 = 1156 - 256 = 900 \]
Taking the square root to find H:
\[ H = \sqrt{900} = 30 \text{ cm} \]

### Step 4: Calculate the volume of the cone
The volume (V) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi R^2 H \]
Substituting the values of R and H:
\[ V = \frac{1}{3} \pi (16^2) (30) \]
Calculating \(16^2\):
\[ 16^2 = 256 \]
Now substituting back into the volume formula:
\[ V = \frac{1}{3} \pi (256) (30) \]
Calculating the product:
\[ V = \frac{1}{3} \pi (7680) \]
Now dividing by 3:
\[ V = 2560 \pi \text{ cm}^3 \]

### Final Answer
The volume of the cone is \(2560 \pi \text{ cm}^3\).

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