If both the radius of the base and the altitude of a cone is doubled, what is the change to the volume of the cone?
To find the change in volume, we need to compare the original volume of the cone with the new volume after doubling the radius of the base and the altitude.
The formula for the volume of a cone is:
V = (1/3) * π * r^2 * h
where V is the volume, π is pi (approximately 3.14159), r is the radius of the base, and h is the altitude.
Let's assume the original radius of the base is r and the original altitude is h. So the original volume, V1, can be calculated as:
V1 = (1/3) * π * r^2 * h
After doubling the radius and the altitude, the new radius, r2, will be 2r, and the new altitude, h2, will be 2h.
The new volume, V2, can be calculated as:
V2 = (1/3) * π * (2r)^2 * (2h)
= (1/3) * π * 4r^2 * 2h
= (8/3) * π * r^2 * h
To find the change in volume, we can compare V2 with V1:
Change in Volume = V2 - V1
= (8/3) * π * r^2 * h - (1/3) * π * r^2 * h
= (7/3) * π * r^2 * h
So, the change in volume when both the radius of the base and the altitude of a cone are doubled is (7/3) times the original volume.
Please note that this change in volume is true only if both the radius and the altitude are doubled proportionally.