An inverted right cone, half filled with water. I've only the volume of the entire cone to go from.

So the volume of the cone = 1/3 the base area * height

I've been told by a friend that the base area of a half scale cone = 1/4 of the larger cones base area; with the height being half.

So... That would make the water equal to the total volume * (Ab * 1/4) * (H * 1/2)

Vc = 20cm^3

Vw = Vc * 1/4 * 1/2
Vw = 20cm^3/8
Vw = 2.5cm^3

What I don't understand is why the base area of the scaled down cone = 1/4 of the base area of the larger cone.

suppose the radius of the original cone is r

then the base area of the original is πr^2

Now, the radius of your half-size cone is r/2
so the base area
= π(r/2)^2
= πr^2/4
= (1/4) πr^2

Thus the base area of the scaled down cone = 1/4 of the base area of the larger cone.

To understand why the base area of the scaled-down cone is 1/4 of the larger cone, we need to consider the relationship between the volumes and the dimensions of the cones.

A cone is a three-dimensional shape with a circular base and a pointed top called the apex. The volume of a cone is given by the formula V = (1/3) * base area * height, where the base area is the area of the circular base and the height is the distance from the apex to the base.

In the case of a half-filled inverted cone, we want to find the volume of the water (Vw) when given the total volume of the entire cone (Vc). Let's denote the base area of the larger cone as Ab_l and the height as H_l, and the base area of the scaled-down cone as Ab_s and the height as H_s.

When you say that the base area of the scaled-down cone is 1/4 of the larger cone and the height is half, it means that Ab_s = (1/4) * Ab_l and H_s = (1/2) * H_l.

Now let's substitute these values into the formula for the volume of the cone. The volume of the larger cone (Vc) is given as 1/3 * Ab_l * H_l.

Vc = (1/3) * Ab_l * H_l

For the scaled-down cone, the volume is given as 1/3 * Ab_s * H_s.

Vw = (1/3) * Ab_s * H_s

Now substitute the expressions for Ab_s and H_s.

Vw = (1/3) * (1/4) * Ab_l * (1/2) * H_l
Vw = (1/24) * Ab_l * H_l

So the volume of the water (Vw) is (1/24) of the volume of the larger cone, given that the base area of the scaled-down cone is (1/4) of the base area of the larger cone and the height is half.

I hope this explanation helps clarify the relationship between the base areas of the cones and how it affects the volume of the water.