A right circular cone has a base diameter of 10 and a height of 18. A second right circular cone has a base diameter of 15. If the volume of the second cone is equal to the volume of the first cone, then what is the height of the second cone?
provide complete solution w explanations pls
Find the volume of the first cone.
You know r = 5 and h = 18
You MUST know the formula
for the second cone you know r = 7.5 and you have to find h
sub in r = 7.5 and h in the formula and set it equal to the volume of the first cone.
Solve for h
let me know what you get.
To find the height of the second cone, we can start by finding the volume of both cones and then equating them.
The formula for the volume of a cone is given by V = (1/3) * π * r^2 * h, where V is the volume, π is a constant approximately equal to 3.14159, r is the radius of the base, and h is the height of the cone.
Let's begin by finding the volume of the first cone with base diameter of 10. We need to find the radius first. The radius is half the diameter, so the radius of the first cone is 10/2 = 5.
Now, we can use the formula to calculate the volume of the first cone:
V1 = (1/3) * π * r1^2 * h1
= (1/3) * π * 5^2 * 18
= (1/3) * 3.14159 * 25 * 18
≈ 471.239 cubic units (rounded to 3 decimal places).
Next, let's calculate the volume of the second cone with base diameter of 15. Again, we start by finding the radius of the second cone. The radius is half the diameter, so the radius of the second cone is 15/2 = 7.5.
Using the formula, we can calculate the volume of the second cone:
V2 = (1/3) * π * r2^2 * h2
= (1/3) * π * 7.5^2 * h2
Since the volume of the second cone is equal to the volume of the first cone (V1 = V2), we can set up the following equation:
(1/3) * π * 5^2 * 18 = (1/3) * π * 7.5^2 * h2
Simplifying the equation, we can cancel out the common terms:
5^2 * 18 = 7.5^2 * h2
25 * 18 = 56.25 * h2
450 = 56.25 * h2
To find h2, we divide both sides by 56.25:
h2 = 450 / 56.25
h2 ≈ 8 (rounded to the nearest whole number)
Therefore, the height of the second cone is approximately 8 units.