Twenty seven solid iron spheres, each of radius r and surface area S, are melted to form a sphere with surface area S'. Find the

• (a) Radius r' of the new sphere
• (b) Ratio of S and S'

To solve the problem, we first need to understand the relationship between the volume of the spheres and their surface areas.

Calculating the Volume of One Sphere

The volume \( V \) of a solid sphere is given by the formula:

V = (4/3)πr³

For one sphere with radius \( r \), the volume is:

V = (4/3)πr³

Total Volume of 27 Spheres

For 27 spheres, the total volume \( V_{total} \) is:

V_{total} = 27 × (4/3)πr³ = 36πr³

Finding the Radius of the New Sphere

When these spheres are melted to form a new sphere, the total volume remains the same. Let \( r' \) be the radius of the new sphere. The volume of the new sphere can be expressed as:

V' = (4/3)π(r')³

Setting the total volume equal to the volume of the new sphere gives:

36πr³ = (4/3)π(r')³

We can simplify this equation by canceling \( π \) and multiplying both sides by \( 3 \):

108r³ = 4(r')³

Now, solving for \( r' \):

(r')³ = 27r³

r' = 3r

Surface Area of the New Sphere

The surface area \( S \) of a sphere is given by:

S = 4πr²

For the new sphere with radius \( r' \):

S' = 4π(r')² = 4π(3r)² = 36(4πr²) = 36S

Calculating the Ratio of Surface Areas

The ratio of the original surface area \( S \) to the new surface area \( S' \) is:

Ratio = S/S' = S/(36S) = 1/36

Summary of Results

• (a) The radius of the new sphere \( r' \) is 3r.
• (b) The ratio of the surface areas \( S \) and \( S' \) is 1:36.