Volume of cones practice answers
If you're looking for practice problems on the volume of cones and their solutions, I can create some for you along with the answers. Remember that the formula for the volume of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where V is the volume, r is the radius of the base of the cone, and h is the height of the cone.
Here are a few practice problems for you to try:
1. Find the volume of a cone with a radius of 3 cm and a height of 4 cm.
Answer:
\[ V = \frac{1}{3} \pi (3)^2 (4) \]
\[ V = \frac{1}{3} \pi \cdot 9 \cdot 4 \]
\[ V = 12 \pi \]
\[ V \approx 37.70 \text{ cm}^3 \]
2. A cone has a volume of 150 cm³ and a radius of 5 cm. Find its height.
Answer:
\[ 150 = \frac{1}{3} \pi (5)^2 h \]
\[ 150 = \frac{1}{3} \pi \cdot 25 \cdot h \]
\[ 150 = \frac{25}{3} \pi \cdot h \]
\[ h = \frac{150 \cdot 3}{25 \pi} \]
\[ h \approx \frac{450}{78.54} \]
\[ h \approx 5.73 \text{ cm} \]
3. If a cone has a height of 10 meters and the volume is 261.8 cubic meters, find the radius of the base of the cone.
Answer:
\[ 261.8 = \frac{1}{3} \pi r^2 (10) \]
\[ 261.8 = \frac{10}{3} \pi \cdot r^2 \]
\[ r^2 = \frac{261.8 \cdot 3}{10 \pi} \]
\[ r^2 \approx \frac{785.4}{31.42} \]
\[ r^2 \approx 25 \]
\[ r = 5 \text{ m} \]
4. Calculate the volume of a cone with a diameter of 14 inches and a height of 6 inches.
Answer: First, find the radius by halving the diameter.
\[ r = \frac{14}{2} = 7 \text{ inches} \]
Now calculate the volume.
\[ V = \frac{1}{3} \pi (7)^2 (6) \]
\[ V = \frac{1}{3} \pi \cdot 49 \cdot 6 \]
\[ V = 102 \pi \]
\[ V \approx 320.44 \text{ in}^3 \]
Try solving these problems yourself to practice finding the volume of cones. Once you've tried solving them, compare your answers to the solutions above to check your work.