Final answer:
To find the volume V of a sphere as a function of its surface area S, the radius r is eliminated from the equation. None of the provided options (a, b, c, d) correctly express the volume as a function of the surface area. The final expression V = S^(3/2) / (6 π) or V = S^3/6π after cubing both sides, shows that all given options are incorrect.
Step-by-step explanation:
To express the volume of a sphere as a function of its surface area, we first need to establish the two known formulae for a sphere:
- Volume (V) of a sphere = 4/3 πr^3
- Surface area (S) of a sphere = 4 πr^2
We want to eliminate the radius (r) from the volume formula to make it a function of the surface area. Using the surface area formula, we can solve for r:
r = √(S / (4 π))
Now, we substitute this value of r into the volume formula:
V = 4/3 π(S / (4 π))^(3/2)
After simplifying:
V = (1/6)π * S^(3/2) / (π^(3/2))
Now, we take the π terms out:
V = (1/6) * S^(3/2) / π
Further simplifying, we get:
V = S^(3/2) / (6 π)
To express this with only S to a power, we square both sides of the equation to get rid of the square root:
V^2 = S^3 / (36 π^2)
Take the cube root of both sides:
V = S^3/(36 π^2)^(1/3)
Finally, we can find the correct option from the ones given:
V = S^3/36π^2 implies V = S^3/(36 π^2)^(1/3) which simplifies to V = S^3/6π, none of the given options match this result; therefore, all provided options (a, b, c, d) are incorrect.