Question

20 identical solid hemispheres of radius 3 cm are melted to form a big solid hemisphere. What is the radius (in cm) of the biggest hemisphere formed, if 20% of the solid is wasted during the process?

• A. 3√2
• B. 6
• C. 6√2
• D. 3

Answer 1

Answer:

C. 6√2

Volume of one small hemisphere
$V=\frac{2}{3}\pi(3)^3=\frac{2}{3}\pi(27)=18\pi$

Volume of 20 hemispheres
$20 \times 18\pi = 360\pi$
20% wastage ⇒ 80% usable
$0.8 \times 360\pi = 288\pi$

Let the radius of the big hemisphere be (R).
$\frac{2}{3}\pi R^3 = 288\pi$

Cancel $(\pi):$
$\frac{2}{3}R^3 = 288$
$R^3 = 432$

Cube root
$432 = 216 \times 2$
$R = \sqrt[3]{432} = \sqrt[3]{216 \times 2} = 6\sqrt[3]{2}$