Cups A and B are cone shaped and have heights of #28 cm# and #23 cm# and openings with radii of #11 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?

Question

Cups A and B are cone shaped and have heights of #28 cm# and #23 cm# and openings with radii of #11 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?

Claire Akin · Accepted Answer

Cup A will be filled upto 15.3967 cm

Volume of cone B # V_b= (1/3) pi r_2^2 h_2# Given #r_2 = 9 cm, h_2 = 23 cm# #V_b = (1/3) * pi * 9^2 * 23 = 1950.929 pi# #cm^3# Similarly Volume of cone A # V_a = (1/3) pi r_1^2 h_1# Given #r-1 = 11 cm, h_1 = 28 cm# #V_a = (1/3) * pi * 11^2 * 37 = 3547.9053 pi# #cm^3# As volume of cone A is greater than the volume of cone B, cup A will not overflow. Volume of partly filled cone A #V_p = (1/3) pi r_1^2 h_3# #V_p = = V_b# # (1/3) pi * 9^2 * 23 = (1/3) * pi * 11^2 * h_3# #h_3 = (cancel((1/3) * pi) * 9^2 * 23) / (cancel((1/3) * pi) * 11^2)# #h_3 =( 1863) / 121 = 15.3967# #cm#

Aaliyah Anderson · 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. If the volume of cup B's contents is greater than the remaining volume in cup A after filling it to its brim, then cup A will overflow.

The volume of a cone can be calculated using the formula: $$V = \frac{1}{3} \pi r^2 h$$ where $r$ is the radius of the cone's base and $h$ is the height of the cone.

First, let's calculate the volume of each cup:
- Cup A: $V_A = \frac{1}{3} \pi (11)^2 (28)$
- Cup B: $V_B = \frac{1}{3} \pi (9)^2 (23)$

Then, we need to find the volume of cup B's contents and see if it fits into cup A:
- Volume of cup B's contents = $V_B$

If the volume of cup B's contents is less than or equal to the remaining volume in cup A after filling it to its brim, then cup A will not overflow. Otherwise, cup A will overflow.

To find out how high cup A will be filled, subtract the volume of cup B's contents from the volume of cup A, then divide by the area of cup A's opening to find the height the liquid will reach.

$$h_{\text{filled}} = \frac{V_A - V_B}{\pi \times (11)^2}$$