A right circular cone is inscribed inside a sphere. The right circular cone has the radius of the base of 4 inches and the height is greater than 2 inches. The sphere has a radius of 5 inches. What is the ratio of the volume of the cone to the volume of the sphere? Leave your answer in fractional form.
please help! thanks (:
The cone's height is 6 cm.
1/9
To find the ratio of the volume of the cone to the volume of the sphere, we need to calculate the volumes of both the cone and the sphere.
1. Volume of the cone:
The volume of a right circular cone is given by the formula V_cone = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height.
Given: Radius of the base of the cone (r) = 4 inches
Let's assume the height of the cone is represented by 'h' inches.
Using the formula, the volume of the cone is:
V_cone = (1/3) * π * (4^2) * h
= (1/3) * π * 16 * h
= (16/3) * π * h cubic inches --(1)
Now, let's calculate the height of the cone.
2. Height of the cone:
Since the cone is inscribed inside the sphere, its diameter will be the same as the diameter of the sphere. Also, the height of the cone will be less than the diameter of the sphere.
Given: Radius of the sphere (R) = 5 inches
Diameter of the sphere = 2 * R = 2 * 5 = 10 inches
So, the height of the cone (h) is less than 10 inches (h < 10), and it is greater than 2 inches (h > 2).
Now, let's move on to calculating the volume of the sphere.
3. Volume of the sphere:
The volume of a sphere is given by the formula V_sphere = (4/3) * π * R^3, where R is the radius of the sphere.
Given: Radius of the sphere (R) = 5 inches
Using the formula, the volume of the sphere is:
V_sphere = (4/3) * π * (5^3)
= (4/3) * π * 125
= (500/3) * π cubic inches --(2)
Now, let's substitute equations (1) and (2) into the ratio:
Ratio = V_cone / V_sphere = [(16/3) * π * h] / [(500/3) * π]
The π cancels out:
Ratio = (16/3 * h) / (500/3)
Simplifying further, the 3 cancels out:
Ratio = 16h / 500
Therefore, the ratio of the volume of the cone to the volume of the sphere is (16h / 500).