Cups A and B are cone shaped and have heights of #33 cm# and #37 cm# and openings with radii of #12 cm# and #7 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 #33 cm# and #37 cm# and openings with radii of #12 cm# and #7 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?

Sadie Allen · Accepted Answer

No, it will only be filled #~~73.4%# of the way

The formula for volume of a cone is #V=pir^2 h/3#, where #r# is the radius and #h# is the height of the cone. For Cup A, #r=33# and #h=12#. Plug that into the formula and you get #V=pi33^2 * 12/3# #V=1089pi * 4# #V=4356pi# For Cup B, #r=37# and #h=7#. Plug that into the formula and you get #V=pi37^2 * 7/3# #V=1369pi *7/3# #V=(9583pi)/3# #V~~3194.3pi# As you can see, Cup A has a bigger volume then Cup B, since #4356>3194.3#, therefor Cup A will not overflow when Cup B is pored into it. As for how high the contents will reach in Cup A, you simply divide Cup B's volume by Cup A's volume to get #(4356pi)/(3194.3pi)=9583/13068~~73.4%# I hope I helped!

Autumn Armstrong · 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.

1. Calculate the volume of cup A:
   Volume_A = (1/3)π(r_A)^2 * h_A

2. Calculate the volume of cup B:
   Volume_B = (1/3)π(r_B)^2 * h_B

3. Compare the volumes of cup A and cup B. If the volume of cup B is greater than or equal to the volume of cup A, cup A will not overflow when the contents of cup B are poured into it.

4. If cup A is not overflowed, calculate how high cup A will be filled by subtracting the initial volume of cup A from the total volume of cup B poured into it.

5. Calculate the remaining volume in cup A and use it to find the height the liquid reaches in cup A using the formula for the volume of a cone.