Question

Cups A and B are cone shaped and have heights of `28 cm` and `15 cm` and openings with radii of `7 cm` and `3 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

Cup A won't overflow. The height of the content in cup A:`6*root(3)(10)~=12.927cm`

Explanation:

Finding full volumes of A and B
`> V_A= (S_(baseA)*h_A)/3=(pi*7^2*28)/3=(1372pi)/3 cm^3`
`> V_B=(S_(baseB)*h_B)/3=(pi*3^2*15)/3=45pi" " cm^3`

Since `V_A>V_B`, cup A won't overflow

So `V' (=V_B)`, the partial volume of cup A occupied by the content of cup B, is given by
`>V'=(S_(base"'")*h')/3=(pi(r')^2*h')/3`
But
`>(r')/(h')=r_A/h_A=7/28=1/4` => `r'=(h')/4`
So
`>V'=pi/3*((h')/4)^2*h'=(pi*(h')^3)/48` => `(h')^3=(48V')/pi=(48*45cancel(pi))/cancel(pi)` => `h'=root(3) (3*16*5*9)=3*2root(3)(2*5)=6root(3)10cm~=12.927cm`