2. Every autumn, Reidor's Rakes (RRs) sells industrial-strength rakes in the White Mountains. RRs builds inventory...

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2. Every autumn, Reidor's Rakes (RRs) sells industrial-strength rakes in the White Mountains. RRs builds inventory before the peak leaf season starts. Due to its remoteness, no replen- ishment is possible until spring. RRs autumn demand for rakes is estimated to have the following demand probability density function: fo(x) = {(20 (20x) 0 x 20 otherwise RRs purchases the rakes for $20.00 each and sells them for $110.00 each. Unsold rakes are donated to charity at the end of the season, accruing an estimated additional penalty of $10.00 per rake, due to shipping & handling charges. (a) Determine FD(x) for x > 0. (b) Determine the number of rakes RRs should order to maximize its profit. (c) Plot and label the expected profit function P(Q) for 0 Q 30. In particular, label P(0), P(Q*), P(20), and P(30). (d) Determine the expected number of rakes that will be donated to charity if the optimal quantity is purchased. 3. Consider the previous problem. Suppose that, if RR pays A dollars for advertising, the demand distribution will become D~U[0, 20]. Assuming that RR is risk neutral, let Amax be the maximum amount RR will be willing to pay in advertising. (a) Determine Amax if RR orders the optimal order quantity found in the previous problem. (b) Determine Amax if RR adjusts its order quantity to fit the new distribution. 2. Every autumn, Reidor's Rakes (RRs) sells industrial-strength rakes in the White Mountains. RRs builds inventory before the peak leaf season starts. Due to its remoteness, no replen- ishment is possible until spring. RRs autumn demand for rakes is estimated to have the following demand probability density function: fo(x) = {(20 (20x) 0 x 20 otherwise RRs purchases the rakes for $20.00 each and sells them for $110.00 each. Unsold rakes are donated to charity at the end of the season, accruing an estimated additional penalty of $10.00 per rake, due to shipping & handling charges. (a) Determine FD(x) for x > 0. (b) Determine the number of rakes RRs should order to maximize its profit. (c) Plot and label the expected profit function P(Q) for 0 Q 30. In particular, label P(0), P(Q*), P(20), and P(30). (d) Determine the expected number of rakes that will be donated to charity if the optimal quantity is purchased. 3. Consider the previous problem. Suppose that, if RR pays A dollars for advertising, the demand distribution will become D~U[0, 20]. Assuming that RR is risk neutral, let Amax be the maximum amount RR will be willing to pay in advertising. (a) Determine Amax if RR orders the optimal order quantity found in the previous problem. (b) Determine Amax if RR adjusts its order quantity to fit the new distribution.