one glass cylinder and one glass cone with the same height and radius

Question

if the height of the glass doubled and then the volumes would increase times ___
(use for formulas to substitute V=1/3πr^2h and V=πr^2h)

Answer 1

First, let's calculate the initial volume of the glass cone and the glass cylinder.

Given that the height and radius of both shapes are the same:
- Volume of the cone (V_cone) = (1/3) * π * r^2 * h
- Volume of the cylinder (V_cylinder) = π * r^2 * h

Since the height and radius are the same for both shapes, let's just use the variable r for the radius and h for the height:

V_cone = (1/3) * π * r^2 * h
V_cylinder = π * r^2 * h

Now, let's calculate the new volume of the glass cone and cylinder when the height is doubled:

New height (2h):
- Volume of the new cone = (1/3) * π * r^2 * 2h
- Volume of the new cylinder = π * r^2 * 2h

Now, let's calculate the increase in volume for both shapes when the height is doubled:

Increase in volume for the cone = (1/3) * π * r^2 * 2h / ((1/3) * π * r^2 * h)
Increase in volume for the cylinder = π * r^2 * 2h / (π * r^2 * h)

Simplify these expressions:

Increase in volume for the cone = 2
Increase in volume for the cylinder = 2

Therefore, when the height of the glass cone and cylinder is doubled, their volumes will increase by a factor of 2.