Fitness Center Plans: Linear Cost Comparison
Two fitness centers advertise different payment plans. Center A charges a $40 joining fee and $25 per month. Center B charges a $100 joining fee and $15 per month. A student plans to join for m months and has a budget of $300 for the total cost. The student's parent will pay the full total upfront. The student wants to know when Center B becomes cheaper than Center A, whether the $300 budget is enough for a 10-month membership at either center, and what to recommend if the family expects to stay for exactly 8 months. The centers charge no extra fees, and the student will not cancel early. Center A offers a free water bottle valued at $5, but the parent says to ignore noncash extras when comparing costs.
1. Write cost functions A(m) and B(m) for m months at each center. Explain what the slope and y-intercept mean in this context.
Read the worked answer
Center A: A(m) = 40 + 25m. Center B: B(m) = 100 + 15m.
For each function, the y-intercept is the joining fee: $40 for A and $100 for B. The slope is the monthly charge: $25 per month for A and $15 per month for B. The slopes show that A costs more each month, while B costs more upfront. The water bottle is not included because the parent says to ignore noncash extras.
2. Solve B(m) < A(m). Interpret the solution in terms of months.
Read the worked answer
Set 100 + 15m < 40 + 25m. Subtract 15m from both sides: 100 < 40 + 10m. Subtract 40: 60 < 10m. Divide by 10: 6 < m, or m > 6. At exactly 6 months the costs are equal: A(6) = 40 + 150 = 190 and B(6) = 100 + 90 = 190. Center B becomes cheaper only after 6 months. For a membership shorter than 6 months, Center A costs less.
3. For an 8-month membership, compare the total costs and recommend a center under the $300 budget. Then check the 10-month case.
Read the worked answer
For 8 months: A(8) = 40 + 25(8) = 240. B(8) = 100 + 15(8) = 220. Both are under $300, and B is cheaper by $20, so recommend Center B for 8 months if the family is sure about the length.
For 10 months: A(10) = 40 + 250 = 290. B(10) = 100 + 150 = 250. Both are under $300, and B is cheaper by $40. The longer the membership, the more B's lower monthly fee outweighs its higher joining fee.