Volume Practice: Comparing and Composite Solids

The volume 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.

The volume of a cylinder is calculated using the formula: $$V = \pi r^2 h$$ , where r is the radius of the base and h is the height.

The volume of a cone is one-third that of a cylinder with the same base and height. Thus, $$V_{cone} = \frac{1}{3} V_{cylinder}$$ .

A composite solid is a three-dimensional shape that is made up of two or more simple shapes, such as cones, cylinders, and prisms.

To find the volume of a composite solid, calculate the volume of each simple shape separately and then sum them up.

The volume of a rectangular prism is calculated using the formula: $$V = l \times w \times h$$ , where l is the length, w is the width, and h is the height.

The volume is calculated as: $$V = 12 \times 7 \times 5 = 420 \text{ cm}^3$$ .

Using the formula: $$V = \frac{1}{3} \pi r^2 h$$ , the volume is: $$V = \frac{1}{3} \pi (3^2)(4) = 12\pi \approx 37.70 \text{ m}^3$$ .

Using the formula: $$V = \pi r^2 h$$ , the volume is: $$V = \pi (3^2)(4) = 36\pi \approx 113.10 \text{ m}^3$$ .

The volume of the cone is one-third of the volume of the cylinder: $$V_{cone} = \frac{1}{3} V_{cylinder}$$ .