Toothbrush manufacturer's have been impregnating their bristles with dye to use as wear indicators. Each bristle is \(0.1 \mathrm{~mm}\) in diameter and \(12 \mathrm{~mm}\) long. Assuming the dye is lost by diffusion from the tip of the bristle into the mouth, i.e., the side wall is impermeable, and the bristle is clear when \(c(t) / c_{o}<0.01 c_{o}\).

a. Determine the concentration profile within the bristle.

b. Assuming continuous brushing and a diffusion coefficient \(D=7 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}\), how long before the toothbrush must be replaced (i.e., half the bristle is clear)?

c. Manufacturer's claims are that the toothbrush will last 3 months. Assuming the average human brushes his/her teeth twice a day for 3 minutes each time, will the toothbrush last that long?