A solid cone is cut into two parts by a plane parallel to the base and 3/5 of the height of the cone above the base. Find the ratio of the volumes in the two parts.
V1/V2 = (pi*r^2*3h/5)/(pi*r^2*2h/5 =
3h/5 * 5/2h = 15h/10h = 3/2.
To find the ratio of the volumes of the two parts, let's first understand the problem.
A solid cone cut into two parts by a plane parallel to the base means that the cone is divided into two sections. One section will be the smaller cone at the top, and the other section will be the remaining bottom part.
We are given that the plane divides the cone at a height that is 3/5 of the total height of the cone.
To find the ratio of the volumes of the two parts, we'll use the concept that the ratio of the volumes of two similar shapes is equal to the cube of their corresponding linear dimensions.
Let's call the original cone's height "H" and the radius of its base "r." The volume of a cone is given by the formula V = (1/3) * π * r² * h.
For the bottom part of the cone, the height will be 3/5 of the original height, which means the height of the bottom cone is (3/5) * H. The radius of the base remains the same as the original cone, so it's still "r."
For the top part of the cone, the height will be 2/5 of the original height, which means the height of the top cone is (2/5) * H. The radius of the base remains the same as the original cone, so it's still "r."
Now, we can calculate the volumes of the two parts.
Volume of the bottom cone, V1 = (1/3) * π * r² * (3/5) * H
Volume of the top cone, V2 = (1/3) * π * r² * (2/5) * H
To find the ratio of the volumes, we divide V2 by V1:
Ratio of volumes, V2/V1 = [(1/3) * π * r² * (2/5) * H] / [(1/3) * π * r² * (3/5) * H]
Simplifying the equation further:
Ratio of volumes, V2/V1 = (2/5 * H) / (3/5 * H)
= 2/3
Therefore, the ratio of the volumes of the two parts is 2:3.