The volume of a right circular cone is 5 liters calculate the volume of the two parts into which the cone is divided by a plane parallel to the base one-third of the way down from vertex to the base give your answer with the nearest Me

1 answer

The volume of the smaller cone can be calculated by using the formula for the volume of a cone:

V = (1/3)πr^2h

where r is the radius and h is the height of the cone.

Given that the volume of the original cone is 5 liters, we can calculate the radius and height of the smaller cone using the ratio of their volumes:

V_original / V_smaller = 2:1

5 / V_smaller = 2

V_smaller = 5 / 2 = 2.5 liters

Since the smaller cone is one-third of the way down from the vertex to the base, the height of the smaller cone is h/3. Therefore, the height of the smaller cone is 1/3 * h.

Now, we can plug in the volume of the smaller cone (2.5 liters) and the height (h/3) into the volume formula to calculate the radius of the smaller cone:

2.5 = (1/3)πr^2(h/3)

2.5 = (1/9)πr^2h

2.5 = (1/9) * 5 * h

h = 2.5 * 9 / 5

h = 4.5 cm

Now, we can calculate the radius of the smaller cone using the height:

2.5 = (1/3)πr^2(4.5/3)

2.5 = (1/9)πr^2 * 4.5

2.5 = (1/2)πr^2

r^2 = 5 / π

r = √(5 / π)

r ≈ 1 cm

Therefore, the volume of the smaller cone is approximately 2.5 liters, and the radius is approximately 1 cm.

The volume of the larger cone is given by:

V_larger = V_original - V_smaller

V_larger = 5 - 2.5

V_larger = 2.5 liters

Therefore, the volume of the two parts into which the cone is divided by a plane parallel to the base one-third of the way down from the vertex to the base are approximately 2.5 liters for each part.