Comparing Two Tutoring Pay Plans
A pre-optometry student is comparing two tutoring jobs for a 10-week term. Center A pays a flat $70 per week plus $16 for each hour tutored; there is no signing fee, but the student must complete a free orientation. Center B pays $23 per hour and guarantees 4 paid hours per week even if fewer students sign up, but charges a one-time $30 materials fee. The student can commit at most 10 hours per week to tutoring and expects to be available the same number of hours h each week at either center. Both centers pay only for completed paid hours, except for Center B’s guaranteed minimum. The student also values predictability and wants to know the weekly-hour threshold where Center B’s higher rate overtakes Center A’s weekly base, and what happens at the 10-hour cap.
1. Write a piecewise expression for 10-week net earnings at each center as a function of h, including Center B’s guarantee.
Read the worked answer
For Center A, weekly pay is 70 + 16h, so 10-week net pay is: A(h) = 10(70 + 16h) = 700 + 160h, for 0 ≤ h ≤ 10.
For Center B, if h ≥ 4, weekly pay is 23h, so B(h) = 10(23h) - 30 = 230h - 30. If h < 4, the guarantee pays for 4 hours per week, so B(h) = 10(23 × 4) - 30 = 920 - 30 = 890. Thus: B(h) = 890 for 0 ≤ h < 4 B(h) = 230h - 30 for 4 ≤ h ≤ 10. At h = 4, both pieces give 890, so the function is continuous at the guarantee boundary.
2. Solve for the h where Center B’s net earnings exceed Center A’s. Is that h within the student’s 10-hour limit?
Read the worked answer
For h ≥ 4, set B > A: 230h - 30 > 700 + 160h 70h > 730 h > 10.4286.
This crossover is above the 10-hour cap, so in the h ≥ 4 region Center A is better for every feasible h. For h < 4, B is fixed at 890, while A = 700 + 160h. Solve 890 > 700 + 160h: 190 > 160h h < 1.1875.
So Center B is better only for very small weekly commitments below about 1.19 hours, because the 4-hour guarantee dominates. For any realistic schedule from about 1.19 to 10 hours per week, Center A pays more. At h = 10, for example, A = 2300 and B = 2270.
3. At 10 hours per week, compare net earnings and state how much the weekly cap would need to increase, in whole hours, for Center B to become preferable.
Read the worked answer
At h = 10: A = 700 + 160(10) = 700 + 1600 = $2300. B = 230(10) - 30 = 2300 - 30 = $2270. Center A is ahead by $30 over the 10-week term.
The crossover from the h ≥ 4 formula is h > 10.4286. Since the student can work only whole hours per week, the cap would need to rise to at least 11 hours per week. Check h = 11: A = 700 + 160(11) = 700 + 1760 = $2460. B = 230(11) - 30 = 2530 - 30 = $2500. At 11 hours per week, Center B is ahead by $40, so the minimum whole-hour increase is 1 hour per week.