Question

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?

Answer 1

`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.