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?

1 Answer
Feb 23, 2016

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