Cups A and B are cone shaped and have heights of #32 cm# and #27 cm# and openings with radii of #12 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?

1 Answer
Jan 22, 2018

#31.9# cm high of cup A will be filled.

Explanation:

Height and radius of cup A is #h_a=32 cm , r_a=12# cm

Volume om cone is #1/3*pi*r^2*h :.V_a=1/3*pi*12^2*32# or

#V_a= 4825.49# cubic cm.

Height and radius of cup B is #h_b=27 cm , r_b=13# cm

# :.V_b=1/3*pi*13^2*27 or V_b= 4778.36# cubic cm.

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

The ratio of radius and hight of cup A is

#r/h=12/32 =3/8 #. The ratio of radius and hight of water cone

is #r_w/h_w=3/8 or r_w=(3*h_w)/8#

. The volume of water cone is #V_w=4778.36# cubic cm.

#:.1/3*pi*r_w^2*h_w =4778.36# or

# 1/3*pi*((3*h_w)/8)^2*h_w =4778.36# or

#h_w^3= (4778.36*64)/(3*pi)~~32447.9835# or

#h_w= root(3)32447.9835 ~~ 31.90(2dp)# cm

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