finding the original height of a truncated cone?

Draw the cross-section of the original cone, and indicate the line where it is cut. You will have a isoceles triangle with a line on the top.

Let R be the radius of full cone, H it's height, r radius of smaller cone, h it's height.

Now, the length of the main base = 2R = 24 cm

The height of the lower piece (bottom) = H-h = 15 cm (given)

Length of smaller triangle base = 2r = 8 cm

Since the bigger triangle and smaller triangle are similar, it's sides and height are proportional:

H/2R = h/2r

H/24 = h/8

Let's substitute for h in terms of H from the equation above:

h = H - 15

H/24 = (H-15)/8

8H = 24H - 24*15

24H-8H = 16H = 24*15

H = 24*15/16 = 3*15/2 = 45/2 = 22.5 cm <======== ANS

Truncated Cone Volume Calculator

When projected from the side, a cone looks like a triangle.

The are 2 similar triangles. The top small triangle and the bigger triangle with the 2 pieces.

Let x be the height of the smaller triangle.

Then:

x / 4 = x+15 / 12 (ratio of sides)

3x / 12 = (x+15) / 12

3x = x +15

2x = 15

x = 7.5 (height of smaller triangle)

Height of the cone is then 7.5 + 15 = 22.5 cm.

22.5 cm