Question

Cups A and B are cone shaped and have heights of `25 cm` and `27 cm` and openings with radii of `8 cm` and `6 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

`21.173\ cm` high from apex.

Explanation:

The volume (`V_A`) of cone cup-A with vertical height `25` cm & radius `8` cm is

`V_A=1/3\pi r^2h=1/3\pi(8^2)(25)=1600/3\pi\ cm^3`

The volume (`V_B`) of cone cup-B with vertical height `27` cm & radius `6` cm is

`V_B=1/3\pi r^2h=1/3\pi(6^2)(27)=324\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{8}{25}`

`r={8}/{25}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=324\pi`

`r^2h=972`

`({8}/{25}h)^2h=972`

`h^3=\frac{972\times 25^2}{8^2}`

`h=\root[3]{\frac{972\times 25^2}{8^2}}`

`h=21.173\ cm`

Thus, the cone cup A will be filled to a vertical height `21.173\ cm` from apex.