Cups A and B are cone shaped and have heights of #20 cm# and #12 cm# and openings with radii of #5 cm# and #3 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 #20 cm# and #12 cm# and openings with radii of #5 cm# and #3 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?

Mason Adams · Accepted Answer

cup A will be filled to a height of #4.32 cm# Volume of cup #A=pir^2h# #=pi(5)^2(20)# #=500pi# Volume of cup #B=pir^2h# #=pi(3)^2(12)# #=108pi# So #pir^2H=108pi# or #r^2H=108# or #5^2H=108# or #25H=108# or #H=108/25# #H=4.32# So cup A will be filled to a height of #4.32 cm#

James Atkins · Answer

undefined To determine if cup A will overflow when the contents of cup B are poured into it, we need to 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 of the base and $ h $ is the height.

Calculate the volumes of cups A and B:
- Volume of cup A: $ V_A = \frac{1}{3} \pi (5^2)(20) $
- Volume of cup B: $ V_B = \frac{1}{3} \pi (3^2)(12) $

Compare the volumes of the two cups. If $ V_B $ is less than or equal to $ V_A $, then cup A will not overflow when the contents of cup B are poured into it. Otherwise, if $ V_B $ is greater than $ V_A $, then cup A will overflow.

Calculate $ V_A $ and $ V_B $:
- $ V_A = \frac{1}{3} \pi (25)(20) = \frac{500}{3} \pi $ cubic cm
- $ V_B = \frac{1}{3} \pi (9)(12) = 36 \pi $ cubic cm

Since $ V_B = 36 \pi $ is less than $ V_A = \frac{500}{3} \pi $, cup A will not overflow when the contents of cup B are poured into it.

To find out how high cup A will be filled, subtract the volume of cup B from the total volume of cup A. Then, divide by the base area of cup A (which is $ \pi (5^2) $) to find the additional height added to cup A.

Calculate the additional height:
- Additional height = $ \frac{V_A - V_B}{\pi (5^2)} $
- $ \frac{\frac{500}{3} \pi - 36 \pi}{25 \pi} = \frac{164}{3} $ cm

So, cup A will be filled with an additional height of $ \frac{164}{3} $ cm when the contents of cup B are poured into it.