Cups A and B are cone shaped and have heights of #35 cm# and #29 cm# and openings with radii of #12 cm# and #16 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 #35 cm# and #29 cm# and openings with radii of #12 cm# and #16 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?

Aaliyah Anderson · Accepted Answer

#=31.42 cm# Volume of Cup #A=pir^2(h/3)# where #r=12 cm# and #h=35 cm# #=pi(12^2)(35/3)=1680pi# Volume of Cup #B=pir^2(h/3)# where #r=16 cm# and #h=29 cm# #=pi(16^2)(29/3)=1508pi# Since volume of #B# is less than volume of #A# it will not overflow Now we can write #pi(12)^2(H/3)=1508pi# or #H=4524/144=31.42 cm# where #H# is the height of Cup #A# will be filled.

Mason Adams · 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 cones.

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.

For cup A:
Radius, $ r = 12 $ cm
Height, $ h = 35 $ cm

For cup B:
Radius, $ r = 16 $ cm
Height, $ h = 29 $ cm

Let's calculate the volumes of the two cones:

For cup A:
$$ V_A = \frac{1}{3} \pi (12^2) (35) $$

For cup B:
$$ V_B = \frac{1}{3} \pi (16^2) (29) $$

After finding the volumes, we can compare them to see if cup A can hold the contents of cup B without overflowing. If cup A's volume is greater than or equal to cup B's volume, then cup A will not overflow. Otherwise, it will overflow.