Cups A and B are cone shaped and have heights of #35 cm# and #21 cm# and openings with radii of #12 cm# and #11 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

#27.856# cm from apex of cup A

Explanation:

The volume (#V_A#) of cone cup-A with vertical height #35# cm & radius #12# cm is

#V_A=1/3\pi r^2h=1/3\pi(12^2)(35)=1680\pi\ cm^3#

The volume (#V_B#) of cone cup-B with vertical height #21# cm & radius #11# cm is

#V_B=1/3\pi r^2h=1/3\pi(11^2)(21)=847\pi\ cm^3#

Since, the volume of cone cup A is more than that of cone cup B hence when content of full cup B is poured into cup A, cup A wouldn't overflow.

Let #h# be the vertical height from apex up to which cup A is filled when content of full cup B is poured. If #r# is the radius of circular surface of content of cup A then using properties of similar triangles

#\frac{r}{h}=\frac{12}{35}#

#r={12}/{35}h#

Now, the volume filled in cone cup A will be equal to the volume of full cone cup B hence we have

#1/3\pir^2h=847\pi#

#r^2h=2541#

#({12}/{35}h)^2h=2541#

#h^3=\frac{2541\times 35^2}{12^2}#

#h=\root[3]{\frac{2541\times 35^2}{12^2}}#

#h=27.856\ cm#

Thus, the cone cup A will be filled to a vertical height #27.856# cm from apex.