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, we can use the formula:

\[ V = \frac{1}{3} \pi r^2 h \]

where:
- \( V \) is the volume,
- \( r \) is the radius of the base,
- \( h \) is the height of the cone,
- \( \pi \) is a constant approximately equal to \( \frac{22}{7} \).

**Step 1: Identify the given values.**
- Radius \( r = 35 \) cm
- Slant height \( l = 37 \) cm

**Step 2: Calculate the height of the cone.**
To find the height \( h \) of the cone, we can use the Pythagorean theorem since the radius, height, and slant height form a right triangle.

\[ l^2 = r^2 + h^2 \]

Substituting the known values:

\[ 37^2 = 35^2 + h^2 \]

Calculating the squares:

\[ 1369 = 1225 + h^2 \]

Now, solve for \( h^2 \):

\[ h^2 = 1369 - 1225 = 144 \]

Taking the square root to find \( h \):

\[ h = \sqrt{144} = 12 \text{ cm} \]

**Step 3: Substitute the values into the volume formula.**

Now we can substitute \( r = 35 \) cm and \( h = 12 \) cm into the volume formula:

\[ V = \frac{1}{3} \pi (35)^2 (12) \]

Calculating \( (35)^2 \):

\[ (35)^2 = 1225 \]

Now substituting back into the volume formula:

\[ V = \frac{1}{3} \times \frac{22}{7} \times 1225 \times 12 \]

**Step 4: Simplify the expression.**

Calculating \( \frac{1}{3} \times \frac{22}{7} \):

\[ \frac{22}{21} \]

Now substituting this back:

\[ V = \frac{22}{21} \times 1225 \times 12 \]

Calculating \( 1225 \times 12 \):

\[ 1225 \times 12 = 14700 \]

Now substituting this value back:

\[ V = \frac{22}{21} \times 14700 \]

**Step 5: Calculate the final volume.**

Now, divide \( 14700 \) by \( 21 \):

\[ 14700 \div 21 = 700 \]

Now multiply by \( 22 \):

\[ V = 700 \times 22 = 15400 \text{ cm}^3 \]

Thus, the volume of the cone is:

\[ \boxed{15400 \text{ cm}^3} \]

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