Cups A and B are cone shaped and have heights of 35 cm35cm and 21 cm21cm and openings with radii of 12 cm12cm and 11 cm11cm, 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

27.85627.856 cm from apex of cup A

Explanation:

The volume (V_AVA) of cone cup-A with vertical height 3535 cm & radius 1212 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.