Cups A and B are cone shaped and have heights of 25 cm25cm and 27 cm27cm and openings with radii of 8 cm8cm and 6 cm6cm, 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

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.