Question

Cups A and B are cone shaped and have heights of `25 cm` and `18 cm` and openings with radii of `8 cm` and `9 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

The cup will not overflow and `24.24` cm high of cup A will be filled.

Explanation:

Height and radius of cup A is `h_a=25 cm , r_a=8` cm

Volume of cup A is `1/3*pi*r^2*h :.V_a=1/3*pi*8^2*25` or

`V_a~~ 1675.52` cubic cm.

Height and radius of cup B is `h_b=18` cm ,`r_b=9` cm

`:.V_b=1/3*pi*9^2*18 or V_b~~ 1526.81` cubic cm

Since `V_a > V_b` , the content will not overflow.

The ratio of radius and height of cup A is `r/h=8/25`.

The ratio of radius and height of content cone in cup A

is `r_c/h_c=8/25 or r_c=(8*h_c)/25`

The volume of content cone is `V_c~~1526.81` cubic cm.

`:.1/3*pi*r_c^2*h_c ~~1526.81` or

`1/3*pi*((8*h_c)/25)^2*h_c ~~1526.81` or

`h_c^3= (1526.81*75)/(8*pi)~~14238.24` or

`h_w= root(3)14238.24 ~~ 24.237 (2dp)` cm

`24.24` cm high of cup A will be filled [Ans]