Cups A and B are cone shaped and have heights of #35 cm# and #29 cm# and openings with radii of #14 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 #14 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?

Savannah Aguilar · Accepted Answer

cup A will overflow.

We need to calculate the volumes of both cups to begin with. The volume (V) of a cone is found using the following formula. #color(red)(|bar(ul(color(white)(a/a)color(black)(V=1/3pir^2h)color(white)(a/a)|)))# where r is the radius and h the height of the cone. cup A has r = 14 and h = 35 #rArrV_A=1/3pixx14^2xx35≈7183.78" cubic cm"# cup B has r = 16 and h = 29 #rArrV_B=1/3pixx16^2xx29≈7774.39" cubic cm"# volume of B > volume of A (7774.39 > 7183.78) and so when cup B is poured into cup A , it will overflow.

Caroline Aucoin · Answer

undefined No, cup A will not overflow. When the contents of cup B are poured into cup A, the volume of cup B's contents will fit entirely within cup A. The height to which cup A will be filled can be calculated using the formula for the volume of a cone. The volume of cup B's contents is equal to the volume of cup A when it's filled to a certain height. Using the formula V = (1/3) * π * r^2 * h, where V is the volume, r is the radius, and h is the height of the cone, you can solve for the height of cup A when filled with the contents of cup B. Plugging in the values for cup A and cup B, the height to which cup A will be filled is approximately 31.25 cm.

Sophia Alfred · Answer

undefined No, cup A will not overflow. When the contents of cup B are poured into cup A, cup A will be filled to a height of $ \frac{{\pi \times 16^2 \times 29}}{{\pi \times 14^2}} = \frac{{16^2 \times 29}}{{14^2}} $ cm, which simplifies to approximately 71.68 cm.