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?

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Answer 1 · 2018-04-29T12:44:59

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

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

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

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

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

Cups A and B are cone shaped and have heights of 37 cm37 cm and 27 cm27 cm and openings with radii of 9 cm9 cm and 5 cm5 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?