Question : Find the total surface area of a sphere whose volume is $\frac{256}{3} \pi \;\text{cm}^3$.
Option 1: $48 \pi \;\text{cm}^2$
Option 2: $64 \pi \;\text{cm}^2$
Option 3: $38 \pi \;\text{cm}^2$
Option 4: $56 \pi \;\text{cm}^2$
Correct Answer: $64 \pi \;\text{cm}^2$
Solution :
The volume of a sphere is $V = \frac{4}{3} \pi r^3$ where V is the volume and r is the radius of the sphere.
Given that the volume of the sphere is $\frac{256}{3} \pi \;\text{cm}^3$.
$⇒\frac{256}{3} \pi = \frac{4}{3} \pi r^3$
$⇒r = \sqrt[3]{\frac{256}{4}} = 4 \;\text{cm}$
The total surface area of a sphere is $A = 4 \pi r^2$.
$⇒A = 4 \pi (4)^2 = 64 \pi \;\text{cm}^2$
Hence, the correct answer is $64 \pi \;\text{cm}^2$.
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