A high-altitude spherical weather balloon expands as it rises, due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.16 inches per second and that r = 38 inches at time t = 0. Determine the equation that models the volume V of the balloon at time t, and find the volume when t = 280 seconds

Question

Mathematics

Alvaro Cox

9 May, 06:22

A high-altitude spherical weather balloon expands as it rises, due to the drop in atmospheric pressure. Suppose that the radius r increases at the rate of 0.16 inches per second and that r = 38 inches at time t = 0. Determine the equation that models the volume V of the balloon at time t, and find the volume when t = 280 seconds

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Rock · Answer 1

Volume of a sphere = (4/3) πr³

At time 0, the radius of the sphere is 36 inches & each second after 0 the radius increases by 0.07 inches, so at any time t our radius is 36 + 0.07t

Therefore at any time t our volume is (4/3) π (36 + 0.07t) ³ so your equation is

V = (4/3) π (36 + 0.07t) ³

At t = 400, this becomes V = (4/3) π (36 + 0.07t) ³ = (4/3) π (36 + 28) ³ (4/3) π (64) ³ = 1098066 cubic inches = 635.4 cubic feet