Cups A and B are cone shaped and have heights of #25 cm# and #18 cm# and openings with radii of #8 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?

1 Answer
Jun 22, 2018

The cup will not overflow and #24.24# cm high of cup A will be filled.

Explanation:

Height and radius of cup A is #h_a=25 cm , r_a=8# cm

Volume of cup A is #1/3*pi*r^2*h :.V_a=1/3*pi*8^2*25# or

#V_a~~ 1675.52# cubic cm.

Height and radius of cup B is #h_b=18# cm ,# r_b=9# cm

# :.V_b=1/3*pi*9^2*18 or V_b~~ 1526.81# cubic cm

Since #V_a > V_b # , the content will not overflow.

The ratio of radius and height of cup A is #r/h=8/25 #.

The ratio of radius and height of content cone in cup A

is #r_c/h_c=8/25 or r_c=(8*h_c)/25#

The volume of content cone is #V_c~~1526.81# cubic cm.

#:.1/3*pi*r_c^2*h_c ~~1526.81# or

# 1/3*pi*((8*h_c)/25)^2*h_c ~~1526.81# or

#h_c^3= (1526.81*75)/(8*pi)~~14238.24# or

#h_w= root(3)14238.24 ~~ 24.237 (2dp)# cm

#24.24# cm high of cup A will be filled [Ans]