Cups A and B are cone shaped and have heights of $32 c m$ $32 cm$ and $27 c m$ $27 cm$ and openings with radii of $12 c m$ $12 cm$ and $12 c m$ $12 cm$ , 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 · 2018-05-24T13:26:52

$27 c m$ $color(blue)(27cm)$

Volume of a cone is given as:

$\frac{1}{3} area of base \times height$ $1/3"area of base"xx "height"$

We can see that both have the same radii, but A has a greater height, so A has a greater volume than B.

Volume of A:

$\frac{1}{3} π (144) (32) = 1536 π$ $1/3pi(144)(32)=1536pi$

Volume of B:

$\frac{1}{3} π (144) (27) = 1296 π$ $1/3pi(144)(27)=1296pi$

Volume of partly filled A is equal to the volume of B:

$\frac{1}{3} π (144) \cdot h = 1296 π$ $1/3pi(144)*h=1296pi$

$h = \frac{3 \cdot 1296 π}{144 π} = 27 c m$ $h=(3*1296pi)/(144pi)=27cm$

Well that's no surprise, since they both have the same radius. I would think this part of the question is incorrect.

Cups A and B are cone shaped and have heights of 32 cm32cm32 cm and 27 cm27cm27 cm and openings with radii of 12 cm12cm12 cm and 12 cm12cm12 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?