A hemispherical bowl is 176 cm round the brim. Supposing it to be half full, how many people may be served from it in hemispherical glasses 4 cm in diameter at the top?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 1372
The number of people served is the liquid in the bowl divided by the volume of one glass. Both are hemispheres, of volume \(\frac{2}{3}\pi r^{3}\); the brim of 176 cm is the circumference of the bowl, which gives its radius.
The bowl (circumference 176 cm):
\(2\pi R=176\), so \(\displaystyle R=\frac{176\times7}{2\times22}=28\) cm
Half of the bowl’s volume \(\displaystyle =\frac{1}{2}\times\frac{2}{3}\pi\times28^{3}=\frac{1}{3}\pi\times21952\)
One glass (diameter 4 cm, so radius 2 cm):
Volume of one glass \(\displaystyle =\frac{2}{3}\pi\times2^{3}=\frac{2}{3}\pi\times8\)
Number of glasses:
Glasses served \(\displaystyle =\frac{\frac{1}{3}\pi\times21952}{\frac{2}{3}\pi\times8}=\frac{21952}{16}=1372\)
Shortcut:
Both are hemispheres, so their volumes are in the ratio of the cubes of the radii: \(\left(\frac{28}{2}\right)^{3}=14^{3}=2744\) glasses fill the whole bowl
The bowl is half full: \(2744\div2=1372\)