A concrete shaft has a slant height of 15 cm and an inside radius of 8 cm. Calculate the volume of the shaft using the inside radius.

Question 2.1: Concrete Shaft

First, we need to find the height ($h$) of the cone. The problem states the slant height is $15 \text{ cm}$. The diagram shows a radius of $8 \text{ cm}$ associated with this slant height. The problem also states the inside radius is $8 \text{ cm}$. Therefore, we will use the inside radius and the given slant height to find the common height of the shaft.

Using the Pythagorean theorem ($h^2 + r^2 = l^2$):

$$h^2 + (8 \text{ cm})^2 = (15 \text{ cm})^2$$ $$h^2 + 64 \text{ cm}^2 = 225 \text{ cm}^2$$ $$h^2 = 225 \text{ cm}^2 - 64 \text{ cm}^2$$ $$h^2 = 161 \text{ cm}^2$$ $$h = \sqrt{161} \text{ cm}$$

The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$.

2.1.1 Calculate the volume of the shaft using the inside radius

Step 1: Identify the inside radius and height.
Inside radius $r_{in} = 8 \text{ cm}$.
Height $h = \sqrt{161} \text{ cm}$.

Step 2: Substitute the values into the volume formula.

$$V_{in} = \frac{1}{3} \pi (8 \text{ cm})^2 (\sqrt{161} \text{ cm})$$ $$V_{in} = \frac{1}{3} \pi (64) \sqrt{161} \text{ cm}^3$$ $$V_{in} = \frac{64\sqrt{161}}{3} \pi \text{ cm}^3$$

Step 3: Calculate the numerical value.

$$V_{in} \approx \frac{64 \times 12.688577 \times 3.14159265}{3} \text{ cm}^3$$ $$$$