To find the ratio of surface area to volume for a sphere, we can use the formulas for surface area and volume of a sphere.
The surface area (SA) of a sphere is given by the formula:
SA = 4πr²
And the volume (V) of a sphere is given by:
V = (4/3)πr³
Now, to find the ratio of the surface area to volume, we can set up the following ratio:
Ratio = SA / V = (4πr²) / ((4/3)πr³)
Now, we can simplify this:
Ratio = (4πr²) * (3 / (4πr³))
Ratio = (3 * 4πr²) / (4πr³)
Ratio = 3 / r
This means that the ratio of the surface area to the volume of a sphere is inversely proportional to the radius of the sphere. Therefore, as the radius increases, this ratio decreases.
So, providing the measurements of the radius will allow us to calculate this specific ratio for that sphere. For example, if the radius is 2 units, the ratio would be:
Ratio = 3 / 2 = 1.5
This gives you a clear understanding of how to derive the ratio of surface area to volume for any sphere based on its radius.