Question

The formula for the surface area of a cone is given by the formula 𝐴 = 𝜋𝑟2 + 𝜋𝑟√𝑟2 + ℎ2 , where A represents the surface area, r represents the radius, and h represents the height. If the surface area of a cone is 225 cm2 and the radius is 4 cm, find the height of the cone.

Answer 1

We are given the formula for the surface area of a cone as A = πr^2 + πr√(r^2 + h^2).

First, we substitute the given values into the formula:
225 = π(4^2) + π(4)√(4^2 + h^2)

Simplifying, we get:
225 = 16π + 4π√(16 + h^2)

Dividing both sides by π, we have:
225/π = 16 + 4√(16 + h^2)

Next, let's solve for √(16 + h^2) by isolating it on one side of the equation:
√(16 + h^2) = (225/π - 16)/4

Squaring both sides of the equation and simplifying, we get:
16 + h^2 = ((225/π - 16)/4)^2

Now, let's solving for h^2 by subtracting 16 from both sides of the equation:
h^2 = ((225/π - 16)/4)^2 - 16

Taking the square root of both sides of the equation, we get:
h = √(((225/π - 16)/4)^2 - 16)

Using a calculator to evaluate the expression on the right side of the equation, we find that h ≈ 6.88 cm.

Therefore, the height of the cone is approximately 6.88 cm.