Find the volume of a cone having radius of the base 35 cm and slant height 37 cm.

Answer

To find the volume of a cone with a radius of the base \( r = 35 \) cm and a slant height \( l = 37 \) cm, we will follow these steps:

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

### Step 2: Find the height of the cone.
Since we are given the slant height \( l \) and the radius \( r \), we can find the height \( h \) using the Pythagorean theorem. The relationship is given by:
\[ l^2 = r^2 + h^2 \]
Rearranging this gives:
\[ h^2 = l^2 - r^2 \]
Substituting the values \( l = 37 \) cm and \( r = 35 \) cm:
\[ h^2 = 37^2 - 35^2 \]
Calculating \( 37^2 \) and \( 35^2 \):
\[ h^2 = 1369 - 1225 \]
\[ h^2 = 144 \]
Taking the square root gives:
\[ h = \sqrt{144} = 12 \text{ cm} \]

### Step 3: Substitute the values into the volume formula.
Now that we have \( r = 35 \) cm and \( h = 12 \) cm, we can substitute these values into the volume formula:
\[ V = \frac{1}{3} \pi (35^2) (12) \]
Calculating \( 35^2 \):
\[ 35^2 = 1225 \]
So, substituting this back into the volume formula:
\[ V = \frac{1}{3} \pi (1225) (12) \]

### Step 4: Simplify the expression.
Calculating \( \frac{1}{3} \times 12 = 4 \):
\[ V = 4 \pi (1225) \]
Now, substituting \( \pi \approx \frac{22}{7} \):
\[ V = 4 \times \frac{22}{7} \times 1225 \]

### Step 5: Calculate the final volume.
Calculating \( 4 \times \frac{22}{7} \):
\[ V = \frac{88}{7} \times 1225 \]
Calculating \( 1225 \div 7 = 175 \):
\[ V = 88 \times 175 \]
Calculating \( 88 \times 175 = 15400 \):
\[ V = 15400 \text{ cm}^3 \]

### Final Answer:
The volume of the cone is \( 15400 \text{ cm}^3 \).

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