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?

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?

Claire Akin · Accepted Answer

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#

Jayden Jackson · Answer

undefined To determine if cup A will overflow when the contents of cup B are poured into it, compare the volumes of the two cups.

The volume of a cone is given by the formula: $V = \frac{1}{3} \pi r^2 h$, where $r$ is the radius and $h$ is the height.

For cup A:
$$ V_A = \frac{1}{3} \pi (9)^2 \cdot 37 $$

For cup B:
$$ V_B = \frac{1}{3} \pi (5)^2 \cdot 27 $$

Calculate both volumes to compare. If $V_B$ (the volume of cup B) is less than or equal to $V_A$ (the volume of cup A), then cup A will not overflow when the contents of cup B are poured into it. If $V_B$ is greater than $V_A$, then cup A will overflow.

After determining that cup A will not overflow, to find how high cup A will be filled, subtract the volume of cup B from the volume of cup A, and then divide by the new radius of cup A to find the additional height.