If the area of the base of a cone is 25 pi square units, express the volume of the cone as a function of its height?
The volume of any cone is one-third area of the base times the height
To express the volume of a cone as a function of its height, we need to use the formula for the volume of a cone, which is given by:
V = 1/3 * π * r^2 * h
Here, V represents the volume, π is a mathematical constant (approximately 3.14159), r is the radius of the base of the cone, and h is the height of the cone.
To find the radius of the base, we can use the formula for the area of a circle, which is given by:
A = π * r^2
Since we know that the area of the base of the cone is 25π square units, we can set up the equation:
25π = π * r^2
To solve for r^2, we divide both sides of the equation by π:
25π/π = r^2
25 = r^2
Taking the square root of both sides gives us the radius:
r = √25
r = 5
Now that we know the radius, we can express the volume of the cone as a function of its height:
V = 1/3 * π * (5^2) * h
Simplifying this equation, we get:
V = 1/3 * π * 25 * h
So, the volume of the cone can be expressed as a function of its height (h) as:
V(h) = 25/3 * π * h