Melting and Recasting of Solid Spheres

Let's consider three solid spheres with radii of 3 cm, 4 cm, and 5 cm. These spheres are melted and recasted into a new solid sphere. We need to find the percentage decrease in the surface area of the new sphere compared to the sum of the surface areas of the original spheres.

Calculating Surface Area

To solve this problem, we first need to calculate the surface areas of the original spheres.

The surface area of a sphere is given by the formula:

Surface Area of a Sphere = 4πr²

where r is the radius of the sphere.

Using this formula, we can calculate the surface areas of the original spheres:

Surface Area of Sphere 1 = 4π(3)²
Surface Area of Sphere 2 = 4π(4)²
Surface Area of Sphere 3 = 4π(5)²

Calculating the Total Surface Area

Next, we need to find the sum of the surface areas of the original spheres:

Total Surface Area of Original Spheres = Surface Area of Sphere 1 + Surface Area of Sphere 2 + Surface Area of Sphere 3

Calculating the Surface Area of the New Sphere

After the original spheres are melted and recasted, they form a new solid sphere. The radius of this new sphere can be calculated using the formula for the volume of a sphere:

Volume of a Sphere = 4/3πr³

Since the volumes of the original spheres are conserved during the melting and recasting process, we can equate the sum of their volumes to the volume of the new sphere:

Volume of Sphere 1 + Volume of Sphere 2 + Volume of Sphere 3 = Volume of New Sphere

Using the formula for the volume of a sphere, we can calculate the radius of the new sphere:

r³ = (4/3π(3)³) + (4/3π(4)³) + (4/3π(5)³)

By solving this equation, we can find the radius of the new sphere.

Calculating the Surface Area of the New Sphere

Once we have the radius of the new sphere, we can calculate its surface area using the formula mentioned earlier:

Surface Area of New Sphere = 4πr²

Calculating the Percentage Decrease

Finally, we can calculate the percentage decrease in the surface area by comparing the surface area of the new sphere with the total surface area of the original spheres:

Percentage Decrease = (Total Surface Area of Original Spheres - Surface Area of New Sphere) / (Total Surface Area of Original Spheres) * 100

By substituting the values we have calculated, we can determine the percentage decrease in the surface area.

Conclusion

Therefore, the correct answer is a 28% decrease in the surface area.

Given