Cups A and B are cone shaped and have heights of #28 cm# and #23 cm# and openings with radii of #6 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 #6 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?

Aaliyah Anderson · Accepted Answer

cup A will overflow.

The first step here is to calculate the volume (V) of the 2 cups using #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. • If #V_B>V_A,"then cup A will overflow"# #V_A=1/3xxpixx6^2xx28≈1055.575" cubic cm"# #V_B=1/3xxpixx9^2xx23≈1950.929" cubic cm"# Since 1950.929 > 1055.575 , then cup A will overflow.

Wyatt Anderson · Answer

undefined No, cup A will not overflow. When cup B is poured into cup A, the volume of cup B will be contained within cup A without overflowing. The height to which cup A will be filled can be calculated by finding the volume of cup B and then finding how much that volume will increase the height of cup A. The volume of a cone is given by the formula V = (1/3)πr²h, where r is the radius and h is the height.

Volume of cup A = (1/3)π(6 cm)²(28 cm)
Volume of cup B = (1/3)π(9 cm)²(23 cm)

When the contents of cup B are poured into cup A, the total volume in cup A becomes the sum of the volumes of cup A and cup B.

Height of cup A after pouring = Volume of cup A + Volume of cup B / (π(6 cm)²)

Calculate the volumes and perform the calculation to find the height of cup A after pouring.