20 uniform solid hemispheres of radius 3 cm are melted to form a larger solid hemisphere. If 20% of the solid is wasted during this process, what is the radius (in cm) of the largest hemisphere formed?

एक hemisphere का आयतन:

$V = \frac{2}{3}\pi r^3$

छोटे hemisphere की त्रिज्या = 3 cm

एक छोटे hemisphere का आयतन:

$V = \frac{2}{3}\pi (3)^3$

$= \frac{2}{3}\pi \times 27$

$= 18\pi$

20 hemispheres का कुल आयतन:

$20 \times 18\pi = 360\pi$

20% solid waste हो गया ⇒ केवल 80% बचा

बचा हुआ आयतन:

$360\pi \times \frac{80}{100} = 288\pi$

मान लें बड़े hemisphere की त्रिज्या = R

तो,

$\frac{2}{3}\pi R^3 = 288\pi$

$\frac{2}{3}R^3 = 288$

$R^3 = 288 \times \frac{3}{2}$

$R^3 = 432$

$R^3 = 216 \times 2$

$R = 6\sqrt[3]{2}$

Correct Answer: $6\sqrt[3]{2}$