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?

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Answer 1 · 2018-01-14T11:40:29

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

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]

Cups A and B are cone shaped and have heights of 34 cm34 cm and 23 cm23 cm and openings with radii of 15 cm15 cm and 17 cm17 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?