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 + 5 inches. (Express your answer in terms of π and r.)

Answer 1

The original volume of the ballon, with radius r, is given by
V₀ = (4/3)πr³ in²

If the radius increases to (r+5) because of air input, the new volume is
V₁ = (4/3)π (r+5)³ in²

The difference in volume is the amount of air used to inflate the balloon.
The volume of air is
V = V₁ - V₀
    = (4/3)π [r³ + 3r²(5) + 3r(5²) + 5³ - r³]
    = (4/3)π (15r² + 75r + 125) in³

Answer:  (4/3)π (15r² + 75r + 125)  in³