Cups A and B are cone shaped and have heights of #33 cm# and #29 cm# and openings with radii of #10 cm# and #13 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 #29 cm# and openings with radii of #10 cm# and #13 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?

Theodore Abad · Accepted Answer

Yes, cup A will overflow.

Given: cup A is a cone with #r = 10 cm; h = 33 cm# #" "#cup B is a cone with #r = 13 cm; h = 29 cm# Volume of a circular right cone: #V = 1/3 pi r^2 h# #V_b = 1/3 pi 13^2 * 29 = 4901/3 pi ~~ 5132 " "cm^3# #V_a = 1/3 pi 10^2 * 33 = 1100 pi ~~ 3456 " "cm^3#

Carson Arnold · Answer

undefined No, cup A will not overflow. When cup B's contents are poured into cup A, the volume of cup B will be transferred to cup A. The volume of a cone can be calculated using the formula V = (1/3)πr²h, where V is the volume, r is the radius of the base, and h is the height.

The volume of cup A is (1/3)π(10²)(33) = 3300π cm³, and the volume of cup B is (1/3)π(13²)(29) = 5303π cm³.

So when cup B is poured into cup A, the total volume of liquid in cup A will be 3300π + 5303π = 8603π cm³.

Now, to find the height of the liquid in cup A, we can use the formula for the volume of a cone and solve for height. Rearranging the formula, we get h = (3V) / (πr²), where h is the height of the liquid in cup A, V is the total volume of liquid poured into cup A, and r is the radius of cup A's opening.

Substituting the values, h = (3 * 8603π) / (π * 10²) ≈ 774.27 cm.

Therefore, cup A will be filled up to approximately 774.27 cm high.