Cups A and B are cone shaped and have heights of $33 cm$ and $27 cm$ and openings with radii of $13 cm$ and $8 cm$ , 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 · 2016-03-05T12:05:07

Cup A will not overflow because $V_b < V_a$
Cup A will be filled up to $h=22.3303$ cm

Compute for the volumes of cups A and B first.
The formula for cone

$V=1/3*pi r^2 h$

For cup A
$V_a=1/3*pi*13^2*33=1859pi=5840.220741927$

For cup B
$V_b=1/3*pi*8^2*27=576pi=1809.557368128$

We need 2 equations to solve for unknown height with $V_b$ poured into $V_a$

For cup A, we need the ratio of Radius to Height:
$r_a/h_a=13/33$
and $r_a=13/33h_a$

Using volume of $V_b=576pi$

$V_b=1/3pi *r_a^2*h_a$

$576pi=1/3*pi*(13/33*h_a)^2*h_a$

$(576*3*33^2)/13^2=h_a^3$

$h_a=root3((576*3*33^2)/13^2$

$h_a=22.3303" " "$ cm

God bless...I hope the explanation is useful.

Cups A and B are cone shaped and have heights of 33 cm33 cm and 27 cm27 cm and openings with radii of 13 cm13 cm and 8 cm8 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?