Question

Cups A and B are cone shaped and have heights of `32 cm` and `15 cm` and openings with radii of `9 cm` and `4 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

`14.477\ cm` high from apex

Explanation:

The volume (`V_A`) of cone cup-A with vertical height `32` cm & radius `9` cm is

`V_A=1/3\pi r^2h=1/3\pi(9^2)(32)=864\pi\ cm^3`

The volume (`V_B`) of cone cup-B with vertical height `15` cm & radius `4` cm is

`V_B=1/3\pi r^2h=1/3\pi(4^2)(15)=80\pi\ cm^3`

Since, the volume of cone cup A is more than that of cone cup B hence when the 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 as follows

`\frac{r}{h}=\frac{9}{32}`

`r={9}/{32}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=80\pi`

`r^2h=240`

`({9}/{32}h)^2h=240`

`h^3=\frac{240\times 32^2}{9^2}`

`h=\root[3]{\frac{240\times 32^2}{9^2}}`

`h=14.477\ cm`

Thus, the cone cup A will be filled to a vertical height `14.477\ cm` from apex.