Question

Cups A and B are cone shaped and have heights of `34 cm` and `23 cm` and openings with radii of `15 cm` and `17 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

`32.44` cm high in cup "A" will be filled.

Explanation:

Cup B (Conical): radius and height are `r_b=17 cm ; h_b =23 cm`

Capcaity of Cup B is `V_b=1/3*pi*r_b^2*h_b`

`V_b=1/3*pi*17^2*23 =6960.72` cubic cm

Cup A (Conical): radius and height are `r_a=15 cm ; h_a =34 cm`

Capcaity of Cup A is `V_a=1/3*pi*r_a^2*h_a`

`V_a=1/3*pi*15^2*34 =8011.06` cubic cm

`r_a/h_a=15/34` . Since the capacity of cup B is less than that of

cup A , the content will not overflow. Let the height and radius in

the content cone in cup A be `h_2 and r_2`

`r_2/h_2=15/34 ; r_2= 15/34*h_2 ; V_c=6960.72 ; V_c` is the

volume of contents in cup A `:. 1/3*pi*r_2^2*h_2 =6960.72`

`r_2^2*h_2 =(6960.72*3)/pi` or

`(15/34*h_2)^2*h_2=(6960.72*3)/pi` or

`h^3=(6960.72*3)/pi*34^2/15^2=34150.8`

`:. h=root(3) 34150.8 ~~32.44(2dp)` cm .

`32.44` cm high in cup "A" will be filled. [Ans]