Cups A and B are cone shaped and have heights of #27 cm# and #24 cm# and openings with radii of #7 cm# and #9 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?

The formula for the volume of a cone is:

The Volume of cup A is:

The Volume of cup B is:

If a full Cup B is poured into an empty Cup A, then Cup A will overflow

To determine whether cup A will overflow when the contents of cup B are poured into it, we need to compare the volumes of the two cups. If the volume of cup B's contents is greater than the remaining volume in cup A after pouring, then cup A will overflow. Otherwise, cup A will not overflow.

The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height.

For cup A: Radius (r) = 7 cm Height (h) = 27 cm

For cup B: Radius (r) = 9 cm Height (h) = 24 cm

First, we calculate the volumes of both cups:

Volume of cup A: V_A = (1/3)π(7^2)(27) = 539π cm^3

Volume of cup B: V_B = (1/3)π(9^2)(24) = 1512π cm^3

Since cup B is full, its contents have a volume of 1512π cm^3.

To find out how much cup A will be filled when the contents of cup B are poured into it, we subtract the volume of cup B's contents from the volume of cup A:

Remaining volume in cup A = V_A - V_B = 539π - 1512π = -973π cm^3

Since the result is negative, it means that cup A cannot contain all of cup B's contents. Therefore, cup A will overflow when the contents of cup B are poured into it.