Question

Cups A and B are cone shaped and have heights of `37 cm` and `27 cm` and openings with radii of `9 cm` and `5 cm`, respectively. If cup B is full and its contents are poured into cup A, will cup A overflow? If not how high will cup A be filled?

Answer 1

A is bigger in both dimensions, so will hold the contents of B at height `h` for `V_B = 1/3 pi r_A^2 h` or

#h = {3 V_B}/{pi r_A^2 } = 3 (1/3 pi r_B^2 h_B}/{\pi r_A^2 } = h_B {r_B^2}/r_A^2 = (27) 5^2/9^2= 25/3 #

Explanation:

The volume of a cone of radius `r` and height `h` is given by

`V = 1/3 pi r^2 h`

A is bigger than B in both radius and height, so of course B's volume is less and A will not overflow. We have

`V_A = 1/3 pi (9^2) 37 = 999 pi text{ cm}^3`

`V_B = 1/3 pi (5^2) (27) = 225 pi text{ cm}^3`

The height of A after receiving the contents of B is given by

`V_B = 1/3 pi r_A^2 h`

`h = {3 V_B}/{pi r_A^2 } = {3 cdot 225 pi}/{pi (9^2) } = 25/3`