The volume of a cone is 27 cm^3. What is the volume of a cylinder that shares the same radius and height as the cone
The volume of a cone can be found using the formula:
Vcone = (1/3) * π * r^2 * h
Given that the volume of the cone is 27 cm^3, we can rewrite the formula:
27 = (1/3) * π * r^2 * h
Since the radius and height of the cone are the same as the cylinder, we can assume that r and h are equal for both shapes. Therefore, let's represent both as "x."
27 = (1/3) * π * x^2 * x
Multiplying both sides of the equation by 3 to get rid of the fraction:
81 = π * x^3
Divide both sides of the equation by π:
x^3 = 81 / π
Taking the cube root of both sides of the equation:
x = (81 / π)^(1/3)
Now, we can calculate the volume of the cylinder using the formula:
Vcylinder = π * r^2 * h
Substituting x for both r and h:
Vcylinder = π * x^2 * x
Vcylinder = π * (81 / π)^(1/3)^2 * (81 / π)^(1/3)
Simplifying:
Vcylinder = π * (81^2 / π^2)^(1/3) * (81 / π)^(1/3)
Vcylinder = π * (6561 / π^2)^(1/3) * (81 / π)^(1/3)
Vcylinder = π * 3 * π^(1/3) * 9^(1/3)
Vcylinder = 27 * π^(4/3)
Therefore, the volume of the cylinder is 27 * π^(4/3) cubic cm.