Answer :
To solve this problem, we need to find the radius of a cone given that the volume of the cone is equal to the volume of a sphere with a given diameter. Let's start by finding the volume of the sphere.
The formula to calculate the volume of a sphere is:
[tex]V_{sphere} = \frac{4}{3} \pi r^3[/tex]
where [tex]r[/tex] is the radius of the sphere. Given that the diameter of the sphere is 12 cm, the radius [tex]r[/tex] is half of the diameter:
[tex]r = \frac{12}{2} = 6 \text{ cm}[/tex]
Plug the radius into the formula:
[tex]V_{sphere} = \frac{4}{3} \pi (6)^3[/tex]
[tex]V_{sphere} = \frac{4}{3} \pi (216)[/tex]
[tex]V_{sphere} = 288 \pi \text{ cubic centimeters}[/tex]
Now, since the volume of the cone is equal to the volume of the sphere, the volume of the cone is also [tex]288 \pi[/tex] cubic centimeters.
The formula for the volume of a cone is:
[tex]V_{cone} = \frac{1}{3} \pi r^2 h[/tex]
where [tex]r[/tex] is the radius of the base of the cone and [tex]h[/tex] is the height of the cone. We are given that the height [tex]h[/tex] of the cone is 20 cm. We set the volume equal to the sphere's volume:
[tex]\frac{1}{3} \pi r^2 (20) = 288 \pi[/tex]
Simplify the equation:
[tex]\frac{20}{3} \pi r^2 = 288 \pi[/tex]
Divide both sides by [tex]\pi[/tex] to cancel out [tex]\pi[/tex]:
[tex]\frac{20}{3} r^2 = 288[/tex]
Multiply both sides by 3 to clear the fraction:
[tex]20 r^2 = 864[/tex]
Divide both sides by 20 to solve for [tex]r^2[/tex]:
[tex]r^2 = \frac{864}{20}[/tex]
[tex]r^2 = 43.2[/tex]
Take the square root of both sides to solve for [tex]r[/tex]:
[tex]r = \sqrt{43.2}[/tex]
[tex]r \approx 6.57 \text{ cm}[/tex]
Therefore, the radius of the cone, rounded to two decimal places, is approximately 6.57 cm.
