Answer :
Time taken to inflate the balloon to two-thirds of its maximum volume is 16 minutes.
Given the diameter of the balloon = 55 ft
Let r be the radius of the balloon. Then r = 55/2 = 27.5 ft
Rate of change of radius = 1.5 ft/min.
The maximum volume of the balloon = [tex]\frac{4}{3}\pi r^3[/tex] = [tex]\frac{4}{3}\times3.14\times 27.5^3[/tex]
= 87069.583 [tex]ft^3[/tex]
Two- thirds of the volume = (2/3) x 87069.583 = 58046.389 [tex]ft^3[/tex]
Let R be the radius of the balloon with two-thirds of its maximum volume.
Then, [tex]\frac{4}{3}\pi R^3[/tex] = 58046.389
⇒ [tex]R^3=\frac{3}{4\times3.14}\times 58046.389[/tex] = 13864.583
⇒ [tex]R=13864.583^\frac{1}{3}[/tex]
⇒ R = 24.023 ft
Now time taken to inflate balloon to the two-third of the maximum volume = 24.023/1.5 = 16 minutes approximately.
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