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
To find the volume of the sphere with radius equal to that of a cone, we need to determine the radius of the cone first.
Given that the cone has a slant height of 10 centimeters and a lateral area of 60π square centimeters, we can use these two pieces of information to solve for the radius of the cone.
The lateral area of a cone is given by the formula: Lateral Area = πrℓ, where r is the radius of the base and ℓ is the slant height.
In this case, the lateral area is given as 60π square centimeters and the slant height is given as 10 centimeters. So we can write the equation as:
60π = πrℓ
To isolate the radius, we divide both sides of the equation by πℓ:
60π / (πℓ) = r
Simplifying, we get:
60 / ℓ = r
Now, we can substitute the value of the slant height ℓ, which is 10 centimeters, into the equation to find the radius of the cone:
60 / 10 = r
So, the radius of the cone is 6 centimeters.
Now that we know the radius of the cone, we can proceed to find the volume of the sphere.
The volume of a sphere is given by the formula: V = (4/3)πr³, where r is the radius.
Plugging in the radius of the cone, which is 6 centimeters, into the formula, we get:
V = (4/3)π(6)³
Calculating further:
V = (4/3)π(216)
V = (4/3)(216π)
V = 288π cubic centimeters
So, the volume of the sphere with a radius equal to that of the cone is 288π cubic centimeters.
To find the volume of the sphere with a radius equal to that of the cone, we first need to calculate the radius of the cone.
Given:
Slant height of the cone = 10 cm
Lateral area of the cone = 60π square cm
The lateral area of a cone is given by the formula:
Lateral Area = πrℓ
Where:
r is the radius of the base of the cone,
ℓ is the slant height of the cone.
In this case, we have the lateral area as 60π square cm.
60π = πr(10)
Dividing both sides by π and solving for r:
r(10) = 60
r = 60/10
r = 6 cm
Now that we have the radius of the cone, which is 6 cm, we can find the volume of the sphere with the same radius.
The volume of a sphere is given by the formula:
Volume = (4/3)πr^3
Substituting the radius:
Volume = (4/3)π(6^3)
Volume = (4/3)π(216)
Volume ≈ 288π cubic cm
Therefore, the volume of the sphere with a radius equal to that of the cone is approximately 288π cubic centimeters.