Question

a cone and a cylinder have equal - sized bases with an 8 in. diameter. they both have a height of 6 in. how can you find the volume of both figures without using the two separate formulas?
a find the volume of the cone. multiply the cone’s volume by 2 to find the volume of the cylinder.
b find the volume of the cylinder. divide the cylinder’s volume by 2 to find the volume of the cone.
c find the volume of the cone. multiply the cone’s volume by 8 to find the volume of the cylinder.
d find the volume of the cylinder. since both figures have the same circular base their volumes are equal. so the cone’s volume is the same as the cylinder’s volume.
e find the volume of the cylinder. divide the cylinder’s volume by 3 to find the volume of the cone.

Explanation:

The formula for the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \), and for a cone with the same base (radius \( r \)) and height (\( h \)) is \( V_{cone} = \frac{1}{3}\pi r^2 h \). This means the volume of a cone is \( \frac{1}{3} \) of the volume of a cylinder with the same base and height. So, if we find the cylinder's volume, we divide it by 3 to get the cone's volume (or multiply the cone's volume by 3 to get the cylinder's). Let's analyze the options:

• Option A: Multiplying cone's volume by 2 is incorrect (should be 3).
• Option B: Dividing by 2 is wrong (ratio is 1/3, so divide by 3).
• Option C: Multiplying by 8 is incorrect.
• Option D: Volumes are not equal (cone is 1/3 of cylinder).
• Option E: Correct, as cylinder volume divided by 3 gives cone's volume.

Answer:

E. Find the volume of the cylinder. Divide the cylinder’s volume by 3 to find the volume of the cone.