Question

Two spheres have volumes of 87 cm³ and 647 cm³. If the surface area of the smaller sphere is 167 cm², what is the
surface area of the larger sphere?
64 cm²
96 cm²
128 cm²
256 cm²

1. trucchi
The surface area of the larger sphere is equal to 636.365 square centimeters.

### How to determine the surface area of the sphere by the use of direct variation formulas

In this question we must estimate the surface area of the smaller sphere. By geometry we know that the volume of a sphere is directly proportional to the cube of its radius and the surface area is directly proportional to the square of radius, then the volume to surface area ratio is equal to:
V/A = k · r      (1)
Where:
• k – Proportionality constant
Then, we can derive the following relationship between the two spheres by eliminating the proportionality constant:
V/(A · r) = V’/(A’ · R)     (2)
Where:
• r – Radius of the smaller sphere.
• R – Radius of the larger sphere.
First, we need to determine the radii of the spheres:
R = ∛(3 · V’ / 4π)
R = ∛(3 · 647 / 4π)
R ≈ 5.365 cm
Smaller sphere
r = ∛(3 · V / 4π)
r = ∛(3 · 87 / 4π)
r ≈ 2.749 cm
Lastly, we find the surface area of the larger sphere:
A · r · V’ = A’ · R · V
A’ = (A · r · V’) / (R · V)
A’ = (167 · 2.749 · 647) / (5.365 · 87)
A’ = 636.365 cm²
The surface area of the larger sphere is equal to 636.365 square centimeters.