Understanding Cone Volume Calculation

This problem asks us to find the volume of a right circular cone given its diameter and slant height. We are provided with the diameter of the base as 30 cm and the slant height as 113 cm. We need to use the value of pi as $\frac{22}{7}$.

Calculating Cone Dimensions

First, let's determine the radius (r) and the height (h) of the cone from the given information.

• The diameter of the base is given as 30 cm. The radius is half of the diameter:
  $r = \frac{\text{diameter}}{2} = \frac{30 \text{ cm}}{2} = 15 \text{ cm}$
• The slant height (l) is given as 113 cm.
• In a right circular cone, the radius, height, and slant height are related by the Pythagorean theorem: $l^2 = r^2 + h^2$.
• We need to find the height (h). Rearranging the formula, we get:
  $h^2 = l^2 - r^2$
  $h = \sqrt{l^2 - r^2}$
• Substitute the values of l and r:
  $h = \sqrt{(113 \text{ cm})^2 - (15 \text{ cm})^2}$
  $h = \sqrt{12769 \text{ cm}^2 - 225 \text{ cm}^2}$
  $h = \sqrt{12544 \text{ cm}^2}$
  $h = 112 \text{ cm}$

Calculating Cone Volume

Now that we have the radius (r = 15 cm) and the height (h = 112 cm), we can calculate the volume (V) using the formula for the volume of a cone:

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

Substitute the known values and $\pi = \frac{22}{7}$:

$$V = \frac{1}{3} \times \frac{22}{7} \times (15 \text{ cm})^2 \times (112 \text{ cm})$$

Calculate the terms:

• $(15 \text{ cm})^2 = 225 \text{ cm}^2$
• $V = \frac{1}{3} \times \frac{22}{7} \times 225 \text{ cm}^2 \times 112 \text{ cm}$

Simplify the calculation:

$$V = \frac{1 \times 22 \times 225 \times 112}{3 \times 7} \text{ cm}^3$$

We can simplify by dividing 225 by 3 and 112 by 7:

• $\frac{225}{3} = 75$
• $\frac{112}{7} = 16$

So, the expression becomes:

$$V = 22 \times 75 \times 16 \text{ cm}^3$$

Perform the multiplication:

• $22 \times 75 = 1650$
• $V = 1650 \times 16 \text{ cm}^3$
• $V = 26400 \text{ cm}^3$

Therefore, the volume of the given right circular cone is 26,400 cm³.