Problem:

A cone with a height of 8 m has a curved surface area of 188.4 square meters. Find its volume.

Solution:

To find the volume of the cone, we need to know its radius and height. We are given the height of the cone as 8 m, but we need to find the radius.

Let's assume the radius of the cone is 'r' meters.

Curved Surface Area of a Cone:

The curved surface area of a cone can be calculated using the formula:

CSA = π * r * l

Where CSA is the curved surface area, r is the radius, and l is the slant height of the cone.

Finding the Slant Height:

To find the slant height, we can use the Pythagorean theorem. In a right triangle formed by the height, radius, and slant height, the height is the perpendicular side, the radius is the base, and the slant height is the hypotenuse.

Using the Pythagorean theorem, we have:

height^2 + radius^2 = slant height^2

Plugging in the given values, we have:

8^2 + r^2 = slant height^2

64 + r^2 = slant height^2

Substituting the Slant Height into the Curved Surface Area Formula:

Now that we have the slant height in terms of the radius, we can substitute it into the formula for the curved surface area:

CSA = π * r * l

CSA = π * r * sqrt(64 + r^2)

Given that the curved surface area is 188.4 square meters, we can plug in this value and solve for the radius:

188.4 = π * r * sqrt(64 + r^2)

Squaring both sides to eliminate the square root:

188.4^2 = (π * r)^2 * (64 + r^2)

Solving for the Radius:

Let's simplify the equation and solve for the radius:

35390.56 = (π^2) * (r^2) * (64 + r^2)

35390.56 = 64π^2 * r^2 + (π^2) * (r^4)

Rearranging the equation to quadratic form:

(π^2) * (r^4) + 64π^2 * r^2 - 35390.56 = 0

This is a quadratic equation in terms of r^2. We can solve it using the quadratic formula:

r^2 = [-b ± sqrt(b^2 - 4ac)] / (2a)

Where a = π^2, b = 64π^2, and c = -35390.56.

Solving the quadratic equation gives us two possible values for r^2. We can discard the negative value since the radius cannot be negative:

r^2 ≈ 6.32

Taking the square root of both sides, we find:

r ≈ 2.51

Calculating the Volume:

Now that we have the radius, we can calculate the volume of the cone using the formula:

Volume = (1/3) * π * r^2 * height

Plugging in