Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let f (t) be the fraction of manufacturers that have complied with the regulations within t months. Note that 0 ≤ f (t) ≤ 1. Suppose that the rate at which new companies comply with the regulations is proportional to the fraction of companies who have not yet complied, with constant of proportionality k = .1.

(a) Construct a differential equation satisfied by f (t).

(b) Use Euler’s method with n = 3 to estimate the fraction of companies that comply with the regulations within the first 3 months.

(c) Solve the differential equation in part (a) and compute f (3).

(d) Compare the answers in parts (b) and (c) and approximate the error in using Euler’s method.