Cups A and B are cone shaped and have heights of #32 cm# and #21 cm# and openings with radii of #15 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?

If Cup B is poured into Cup A, the cup will not overflow. To find the height of water in Cup A,

Since the volume of cup B (the smaller cup) is poured into cup A (the larger cup), we can compare their volumes. Cup A has a larger volume than cup B, so pouring the contents of cup B into cup A will not cause cup A to overflow. To find out how high cup A will be filled, we can calculate the additional height the liquid from cup B will add to cup A.

Using the formula for the volume of a cone ( V = \frac{1}{3} \pi r^2 h ), where ( r ) is the radius and ( h ) is the height, we can find the volumes of cup A and cup B. Then, we subtract the original volume of cup A from the combined volume of both cups to find the additional volume that will be added to cup A. Finally, we divide this additional volume by the new base area of cup A to find the additional height the liquid will rise.

Let's denote:

• ( r_A = 15 ) cm (radius of cup A)
• ( h_A = 32 ) cm (height of cup A)
• ( r_B = 9 ) cm (radius of cup B)
• ( h_B = 21 ) cm (height of cup B)

First, calculate the volumes of both cups:

[ V_A = \frac{1}{3} \pi (15)^2 (32) ] [ V_B = \frac{1}{3} \pi (9)^2 (21) ]

Next, calculate the additional volume that will be added to cup A:

[ \text{Additional Volume} = V_B ]

Finally, calculate the additional height cup A will be filled:

[ \text{Additional Height} = \frac{\text{Additional Volume}}{\pi (15)^2} ]

Solve for the additional height to find out how high cup A will be filled when the contents of cup B are poured into it.