Comparing Gym Membership Costs
A fitness center offers two membership plans. Plan A charges a $40 joining fee plus $25 per month. Plan B charges no joining fee but $35 per month. A customer is comparing total costs over m months. The center also offers a coupon that reduces Plan A’s joining fee to $10 if the customer signs up by Friday. The customer plans to stay a member for at least 6 months but is unsure about the exact duration. They want to know when each plan is cheaper and how the coupon changes the decision. The customer also asks what the slope of the cost equation for Plan B represents in this context. They have budgeted $300 for the first year but may cancel earlier.
1. Write an equation for the total cost of each plan without the coupon, using m for months. Then find the number of months for which the two plans cost the same.
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Without the coupon, Plan A costs $40 + $25m, and Plan B costs $35m. Setting them equal: 40 + 25m = 35m → 40 = 10m → m = 4. So after 4 months, both plans cost $140. For m < 4, Plan B is cheaper; for m > 4, Plan A is cheaper. The break-even point is 4 months.
2. With the coupon, which plan is cheaper for 8 months? Show calculations.
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With the coupon, Plan A costs $10 + $25m. For m = 8: Plan A = 10 + 25(8) = 10 + 200 = $210. Plan B = 35(8) = $280. Plan A is cheaper by $70. Even without the coupon, Plan A at 8 months is $40 + 200 = $240, still cheaper than $280, so the coupon strengthens but does not change the decision for 8 months.
3. Interpret the slope of Plan B’s cost equation in context.
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The slope of Plan B's cost equation, C = 35m, is 35. In context, it means the cost increases by $35 for each additional month of membership. The slope represents the monthly fee. The y-intercept is 0, meaning there is no joining fee for Plan B.