Cups A and B are cone shaped and have heights of #34 cm# and #27 cm# and openings with radii of #11 cm# and #5 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 #34 cm# and #27 cm# and openings with radii of #11 cm# and #5 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?

Aurora Ayala · Accepted Answer

Cup A will not overflow. It will be filled up to # color(violet)(5.58 cm)# height

Cone A #r_a = 11 cm, h_a = 34 cm# Cone B #r_b = 5 cm, h_b = 27 cm# Volume of Cone A #V_a = (1/3) pi (r_a)^2 h_a# #V_a = (1/3) pi * 11^2 * 34 = 4308.17 cm^3# Volume of Cone A #V_b = (1/3) pi (r_b)^2 h_b# #V_a = (1/3) pi * 5^2 * 27 = 706.86 cm^3# Since #V_b < V_a#, it will not overflow. #(1/3) pi (r_a)^2 * h = V_b = 225 pi# #h = (3 * 225 * pi) / (pi * 11^2) = 5.58 cm#

Theodore Abad · Answer

undefined To determine whether 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 calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height.

For cup A, with a height of 34 cm and a radius of 11 cm, the volume is V_A = (1/3)π(11^2)(34) = 4182π cm^3.

For cup B, with a height of 27 cm and a radius of 5 cm, the volume is V_B = (1/3)π(5^2)(27) = 175π cm^3.

When the contents of cup B are poured into cup A, the total volume in cup A will be the sum of the original volume of cup A and the volume of cup B.

V_total = V_A + V_B = 4182π + 175π = 4357π cm^3.

To determine how high cup A will be filled, we need to find the height of the liquid in cup A. We can use the formula for the volume of a cone and rearrange it to solve for h:

V = (1/3)πr^2h

h = 3V/(πr^2)

Substituting the total volume V_total and the radius of cup A r_A = 11 cm into the formula, we find:

h_A = 3(4357π)/(π(11^2)) = 111 cm

Therefore, when the contents of cup B are poured into cup A, cup A will not overflow, and it will be filled to a height of 111 cm.