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?

1 Answer

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.