Cups A and B are cone shaped and have heights of 24 cm and 26 cm and openings with radii of 10 cm and 7 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?

Mar 8, 2016

No overflow , h ≈ 12.74cm

Explanation:

Volume of a cone (V) $= \frac{1}{3} \pi {r}^{2} h$
where h represents height and r , the radius.

V_A = 1/3pixx10^2xx24 ≈ 2513.274 " cubic cm "

and V_B = 1/3pixx7^2xx26 ≈ 1334.13 " cubic cm "

There will be no overflow since ${V}_{B} < {V}_{A}$

the volume in cup A will therefore be 1334.13 cubic cm

require to find height with this volume in cup A

$\Rightarrow {V}_{A} = \frac{1}{3} \pi \times {10}^{2} \times h = 1334.13$

multiply both sides by 3 to eliminate fraction

thus: $\pi \times {10}^{2} h = 3 \times 1334.13$

divide both sides by$\left(\pi \times {10}^{2}\right)$

$\Rightarrow \frac{\cancel{\pi \times {10}^{2}} h}{\cancel{\pi \times {10}^{2}}} = \frac{3 \times 1334.13}{\pi \times {10}^{2}}$

the height of liquid in cup A ≈ 12.74cm