Cups A and B are cone shaped and have heights of #32 cm# and #12 cm# and openings with radii of #18 cm# and #6 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 #32 cm# and #12 cm# and openings with radii of #18 cm# and #6 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?

Autumn Armstrong · Accepted Answer

Find the volume of each one and compare them. Then, use cup's A volume on cup B and find the height.

Cup A will not overflow and height will be:

#h_A'=1,bar(333)cm# The volume of a cone: #V=1/3b*h# where #b# is the base and equal to #π*r^2# #h# is the height. Cup A #V_A=1/3b_A*h_A# #V_A=1/3(π*18^2)*32# #V_A=3456πcm^3# Cup B #V_B=1/3b_B*h_B# #V_B=1/3(π*6^2)*12# #V_B=144πcm^3# Since #V_A>V_B# the cup will not overflow. The new liquid volume of cup A after the pouring will be #V_A'=V_B#: #V_A'=1/3b_A*h_A'# #V_B=1/3b_A*h_A'# #h_A'=3(V_B)/b_A# #h_A'=3(144π)/(π*18^2)# #h_A'=1,bar(333)cm#

Aurora Ayala · 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. Since both cups are cone-shaped, we can use the formula for the volume of a cone:

$$ V = \frac{1}{3} \times \pi \times r^2 \times h $$

where:
- $ V $ is the volume of the cone,
- $ r $ is the radius of the cone's base,
- $ h $ is the height of the cone.

First, we calculate the volumes of cups A and B.

For cup A:
$$ V_A = \frac{1}{3} \times \pi \times (18 \, \text{cm})^2 \times 32 \, \text{cm} $$

For cup B:
$$ V_B = \frac{1}{3} \times \pi \times (6 \, \text{cm})^2 \times 12 \, \text{cm} $$

Once we have calculated the volumes of both cups, we compare them. If the volume of cup B is greater than the volume of cup A, then cup A will not overflow when the contents of cup B are poured into it. Otherwise, cup A will overflow.

After calculating the volumes of cups A and B, if cup A does not overflow, we can determine how high cup A will be filled by finding the ratio of the volume of cup B to the volume of cup A and multiplying this ratio by the height of cup A.

Let's perform the calculations to determine if cup A will overflow and, if not, how high it will be filled.