Answer:
The time it would take to inflate the balloon to approximately two-thirds of its maximum volume is approximately 46.90 minutes
Explanation:
The given parameters are;
The diameter of the balloon = 55 feet
The rate of increase of the radius of the balloon when inflated = 1.5 feet per minute
We have;
dr/dt = 1.5 feet per minute = 1.5 ft/min
V = 4/3·π·r³
The maximum volume of the balloon = 4/3 × 3.14 × 55³ = 696556.67 ft³
When the volume is two-thirds the maximum volume, we have;
2/3 × 696556.67 ft³ = 464371.11 ft³
The value of the radius, r₂ at that point is found as follows;
4/3·π·r₂³ = 464371.11 ft³
r₂³ = 464371.11 ft³ × 3/4 = 348278.33 ft³
348278.333333
r₂ = ∛(348278.33 ft³) ≈ 70.36 ft
The time for the radius to increase to the above length = Length/(Rate of increase of length of the radius)
The time for the radius to increase to the above length ≈ 70.369 ft/(1.5 ft/min) ≈ 46.90 minutes
The time it would take to inflate the balloon to approximately two-thirds of its maximum volume ≈ 46.90 minutes.