J & M have their child in daycare twice a week. Being busy people they are often a few minutes late to pick her up. The daycare has a strict policy that parents need to be on time. They enforce this by charging $1 per minute for tardiness.

Suppose that each day the amount of time in minutes that they are late follows an exponential distribution with mean 6.

(a) Their child will be in daycare for 100 days this year. Estimate the probability that they will pay more than $630 in late fees?

(b) The late fees were not effective in getting J & M to arrive on time, so the daycare changed the rate to t ² + t dollars for t minutes of tardiness. On average, how much will J & M pay in late fees each day?

For part (a) we want a numerical answer. For part (b) leave your answer as an integral. Do not compute it out.