Determine the height of a right cone

whose lateral area is 136rr ft^2 and base area is 64rr ft^2

To determine the height of a right cone, it is necessary to have the values for the lateral area and the base area.

The lateral area of a cone is given by the formula A_lateral = π * r * L, where r is the radius of the base and L is the slant height of the cone.

The base area of a cone is given by the formula A_base = π * r^2, where r is the radius of the base.

In this case, we are given that the lateral area is 136π ft^2 and the base area is 64π ft^2.

First, let's find the radius of the base.
From the given base area, we have: A_base = 64π ft^2.
Using the formula for the base area, we can rewrite it as:
64π = π * r^2

To find r, we can divide both sides of the equation by π:
64 = r^2

Taking the square root of both sides, we get:
r = √64 = 8 ft

Now, let's find the slant height (L) using the lateral area.
From the given lateral area, we have: A_lateral = 136π ft^2.
Using the formula for the lateral area:
136π = π * r * L

Substituting the value of the radius (r = 8 ft), we have:
136π = π * 8 * L

Canceling out the π, we have:
136 = 8L

Dividing both sides by 8, we get:
L = 17 ft

Finally, we can use the Pythagorean theorem to find the height (h) of the cone.
In a right cone, the height, slant height, and radius form a right triangle. By applying the Pythagorean theorem:
h^2 = L^2 - r^2
h^2 = 17^2 - 8^2
h^2 = 289 - 64
h^2 = 225

Taking the square root of both sides, we get:
h = √225 = 15 ft

Therefore, the height of the right cone is 15 ft.