Question

A bicycle inner tube can be considered as a joined cylinder of fixed length 200 cm and radius r cm. The radius r increases as the inner tube is pumped up. Air is being pumped into the inner tube so that volume of air in the tube increases at a constant rate of 30cm^3s^-1. Find the rate at which the radius of the inner tube is increasing when r= 2 cm.

Answer 1

This is a problem of application of derivatives.

Consider that the volume of a cylinder is:

V = π·r²·l

next, derivate respect to t:

dV/dt = π·l·(2r)(dr/dt) = 2π·r·l·dr/dt

take into account that dV/dt = 30cm³/s, then, for r=2 cm and l=200cm, you have:

30 cm³/s = 2(3.1415)(2 cm)(200 cm)(dr/dt)

30 cm³/s = 2,513.27 cm² · dr/dt

solve for dr/dt:

dr/dt = (30 cm³/s)/(2,513.27 cm²)

dr/dt = 0.011 cm/s

The last result is the rate at which the radius of the inner tube is increasing