A cone has a slant height of 10 centimeters and lateral area of 60pi square centimeters. What is the volume of the sphere with a radius equal to that of a cone?
-we tried solving and got 4.37 or so for the radius and we couldn't get any further than that
A lateral area=Pi*R*l, where R-radius,
l-slant height
Pi*R*10=60*Pi
R=6
The volume of sphere=(4/3)*Pi*R^3=
1.33*3.14*6^3=902.06
more accurately 1.33333*3.14159*216=904.78
To calculate the volume of the sphere with a radius equal to that of a cone, we first need to find the radius of the cone. From the given information, we know that the slant height of the cone is 10 centimeters.
The slant height, height, and radius of a cone form a right triangle. We can use the Pythagorean theorem to find the height of the cone.
Let's consider the slant height as the hypotenuse, the radius as one side, and the height as the other side of the right triangle. Using the Pythagorean theorem, we have:
(radius)^2 + (height)^2 = (slant height)^2
Let's denote the radius as "r" and the height as "h" for easier calculations. So we have:
r^2 + h^2 = 10^2
r^2 + h^2 = 100
Now, we also know the lateral area of the cone is given as 60π square centimeters. The lateral area of a cone can be calculated using the formula:
lateral area = π * r * slant height
Substituting the given values, we have:
60π = π * r * 10
r = 6
Therefore, the radius of the cone is 6 centimeters.
Now, let's move on to calculating the volume of the sphere with a radius equal to that of the cone. The formula for the volume of a sphere is:
volume = (4/3) * π * (radius)^3
As the cone and sphere have the same radius (6 centimeters), the volume of the sphere is:
volume = (4/3) * π * (6)^3
volume ≈ 904.779 cubic centimeters
So, the volume of the sphere with a radius equal to that of the cone is approximately 904.779 cubic centimeters.