Question

4 A metal sphere having a radius of 10 centimeters is melted down and molded into 8 identical smaller solid spheres. What is the ratio of the surface area of the original sphere to the total surface area of all 8 smaller spheres?

• A. 1 : 2
• B. 2 : 1
• C. 1 : 4
• D. 1 : 1

Answer 1

Answer:

A. 1 : 2

Use the sphere formulas:

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Let the radius of each smaller sphere be (r).

Since the metal is melted and remolded, volume is conserved:

$\frac{4}{3}\pi (10)^3 = 8 \times \frac{4}{3}\pi r^3$

$1000 = 8r^3$

$r^3 = 125 \Rightarrow r = 5 \text{ cm}$

Now use the surface area formula:

• Surface area of the original sphere:
  $4\pi(10)^2 = 400\pi$
  

• Surface area of one small sphere:
  $4\pi(5)^2 = 100\pi$
  

• Total surface area of 8 small spheres:
  $8 \times 100\pi = 800\pi$
  

Therefore, the ratio is:

$400\pi : 800\pi = 1 : 2$