A spherical balloon with radius r inches has volume V(r) = 4 3 πr3. Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches. (Express your answer in terms of π and r.)

Question

A spherical balloon with radius r inches has volume  V(r) =  4 3 πr3.  Find an expression that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 1 inches. (Express your answer in terms of π and r.)

directrix · Answer

From the formula, the volume of air required for inflation of the sphere with radius r is just that, the formula, V = (4/3) π r³.

The amount of air required to inflate a sphere with radius (r + 1) is the same formula with r replaced by (r + 1).  V =  (4/3) π (r + 1)³.

To get the volume of air to inflate from a sphere of radius r to a sphere of radius (r+1), crank out and simplify this difference:

 (4/3) π (r + 1)³ - (4/3) π r³ = ? @saigetyndall