a cone is designed to fit perfectly into a cylinderical container given the volume of the cone is 100 cm3 and the curved surface area of the cylinder is 150 cm2, what is the height of the container?
To find the height of the container, we need to consider the properties of the cone and the cylinder.
Let's start by calculating the height of the cone using the given volume. The formula for the volume of a cone is given by V = 1/3 * π * r^2 * h, where V represents the volume, π represents the mathematical constant pi (approximately 3.14159), r represents the radius of the base of the cone, and h represents the height of the cone.
We are given the volume of the cone as 100 cm^3, so we can write the equation as:
100 = 1/3 * π * r^2 * h
Next, let's find the curved surface area of the cylinder. The formula for the curved surface area of a cylinder is given by A = 2 * π * r * h, where A represents the curved surface area, r represents the radius of the base of the cylinder, and h represents the height of the cylinder.
We are given the curved surface area of the cylinder as 150 cm^2, so we can write the equation as:
150 = 2 * π * r * h
Now, we have two equations with two variables (r and h). We can use these equations to solve for the height of the cylinder.
To eliminate r, we can divide the equation for the volume of the cone by the equation for the curved surface area of the cylinder:
(1/3 * π * r^2 * h) / (2 * π * r * h)
Simplifying the equation, we cancel out the π, r, and h terms:
1/3 * r = 2
Multiplying both sides of the equation by 3, we find:
r = 6
Now that we have found the radius of the base of the cone, we can substitute it back into either of the original equations to solve for h.
Let's substitute r = 6 into the equation for the volume of the cone:
100 = 1/3 * π * (6^2) * h
100 = 36πh
Dividing both sides of the equation by 36π, we find:
h = 100 / (36π)
Using a calculator, we can find the approximate value of h:
h ≈ 0.294 cm
Therefore, the height of the container is approximately 0.294 cm.