A large number of identical items are placed into service at time 0. The items have a failure rate function given by                                         r (t) = 1.105 + 0.30t,where t is measured in years of operation.a. Derive R(t) and F(t).b. If 300 items are still operating at time t = 1 year, approximately how many items would you expect to fail between year 1 and year 2?c. Does the value of r(1) yield a good approximation to the conditional probability computed in part (b)? Why or why not?d. Repeat the calculation of part (b), but determine the expected number of items that fail between t = 1 year and t = 1 year plus 1 week. Does r (t)Δt provide a reasonable approximation to the conditional probability in this case? Why or why not?

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.