Delivery Driver Income Plan: Piecewise Function and Break-Even
A delivery driver is comparing two income plans for the coming month. Under Plan A, she earns a base fee of $3.50 per delivery plus $0.90 per mile driven while carrying an order. She also receives a $50 bonus if she completes more than 120 deliveries in the month. Her fixed monthly expenses are $180 for insurance and $260 for vehicle payment, and she budgets $0.25 per mile for fuel and maintenance. Last month she completed 110 deliveries and averaged 5.2 miles per delivery. She wants to net at least $900 after all expenses. A part-time warehouse job would pay $15.50 per hour for 25 hours per week, but would require her to give up 20 deliveries per week. The driver estimates that each delivery takes 22 minutes, including pickup and drop-off, and that she can work at most 30 hours per week on delivery. She is trying to decide whether to stay with Plan A, increase her delivery count, or take the warehouse job.
1. Write a function for monthly net income under Plan A in terms of number of deliveries d and average miles per delivery m. State the two cases for the bonus.
Read the worked answer
Let d be deliveries and m average miles per delivery. Revenue is 3.50d + 0.90dm, plus a $50 bonus when d > 120. Costs are 180 + 260 + 0.25dm = 440 + 0.25dm. Net for d > 120: N(d,m) = 3.50d + 0.90dm + 50 − 440 − 0.25dm = 3.50d + 0.65dm − 390. For d ≤ 120, the bonus is zero, so N = 3.50d + 0.65dm − 440. The coefficient 0.65 is the net contribution per mile after fuel and maintenance; 3.50 is the net base fee per delivery before miles.
2. Using last month's figures, compute her net income. How far short of $900 is she?
Read the worked answer
At d = 110 and m = 5.2, use the no-bonus formula: 3.50 × 110 = 385; 0.65 × 110 × 5.2 = 0.65 × 572 = 371.80; N = 385 + 371.80 − 440 = $316.80. She is $900 − $316.80 = $583.20 short. If she reaches 121 deliveries at the same average miles, N = 3.50 × 121 + 0.65 × 121 × 5.2 − 390 = 423.50 + 408.98 − 390 = $442.48. The $50 bonus alone does not close the gap; the driver needs substantially more deliveries or longer average trips.
3. At an average of 5.0 miles per delivery, how many deliveries are needed to net $900? Check the time constraint and compare with the warehouse job.
Read the worked answer
Set m = 5.0 and require N ≥ 900 with the bonus: 3.50d + 0.65 × 5.0d − 390 = 3.50d + 3.25d − 390 = 6.75d − 390. Then 6.75d ≥ 1,290 → d ≥ 191.1, so 192 deliveries. Time: 192 × 22 min = 4,224 min = 70.4 hours per month, about 17.6 hours per week, within the 30-hour limit. Net at 192 deliveries = 6.75 × 192 − 390 = $906. The warehouse job pays $15.50 × 25 × 4 = $1,550 per month before expenses, far more for fewer hours. However, it displaces 80 deliveries per month, and the driver should compare non-cash costs such as vehicle wear and schedule flexibility, and verify that the warehouse hours are guaranteed.