Question

The volume of a sphere v=4/3pier^3 well the surface area is S equals 4 pi r ^2 Express the volume of the spear as a function of the surface area

a) V= S^3/36π^2
b) V = S^2/9π
c) V = S^3/12π
d) v = s^2/16π

Answer 1

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.