Choosing a DJ with Linear Cost Models and a Percent Discount
The student council is booking a DJ for the spring dance, which runs from 7:00 to 11:00 p.m., a 4-hour event. Two companies submitted bids. Beat Street charges a $250 booking fee plus $35 per hour. City Spin charges a $120 booking fee plus $55 per hour. The council has raised $400 and cannot exceed it. A parent volunteer says, 'Beat Street is obviously cheaper because its hourly rate is lower.' The council president is not so sure and wants the comparison done with equations, not intuition. City Spin has also offered a 10% discount on the entire bill for events booked for 5 hours or more, which has the council wondering whether extending the dance would change the decision. The venue permit allows the dance to end no later than midnight, so the maximum possible event length is 5 hours. The council must choose a DJ and be able to explain the choice to the full membership.
1. Write an equation for each DJ's total cost for h hours, solve for the duration at which the costs are equal, and interpret the solution, including whether it falls within the permit's limit.
Read the worked answer
Let h be the event length in hours.
Beat Street: A = 250 + 35h City Spin: B = 120 + 55h
Set them equal: 250 + 35h = 120 + 55h. Subtract 35h from both sides: 250 = 120 + 20h. Subtract 120: 130 = 20h, so h = 6.5.
At h = 6.5, A = 250 + 35(6.5) = 250 + 227.50 = $477.50, and B = 120 + 55(6.5) = 120 + 357.50 = $477.50, which checks.
The break-even is 6.5 hours, but the permit caps the event at 5 hours. The solution is outside the feasible domain, so within the allowed range the two cost lines never cross; one DJ is cheaper for every permitted length.
2. For the 4-hour dance, which DJ costs less and by how much? Do both fit the $400 budget? Show the substitutions.
Read the worked answer
Beat Street at 4 hours: 250 + 35(4) = 250 + 140 = $390. City Spin at 4 hours: 120 + 55(4) = 120 + 220 = $340.
City Spin costs $50 less (390 − 340 = 50). Both fit under $400, but Beat Street leaves only $10 of the budget, which is risky if any surcharge appears, while City Spin leaves $60.
Note why the parent's reasoning fails: Beat Street has the lower hourly rate, but its booking fee is $130 higher, and at short durations the fee dominates. The hourly rate alone does not determine total cost.
3. City Spin's 10% discount applies to the entire bill for bookings of 5 hours or more. At the maximum allowed 5 hours, compare the discounted City Spin cost with Beat Street. Does the discount change the decision? At what duration would the discounted costs be equal, and is that duration allowed?
Read the worked answer
City Spin at 5 hours before discount: 120 + 55(5) = 120 + 275 = $395. The discount is 10% off the whole bill: 0.90 × 395 = $355.50. Beat Street at 5 hours: 250 + 35(5) = 250 + 175 = $425. City Spin is cheaper by 425 − 355.50 = $69.50.
Equal-cost duration with the discount: 250 + 35h = 0.90(120 + 55h) = 108 + 49.50h. Then 142 = 14.50h, so h ≈ 9.79 hours.
That break-even is far beyond the 5-hour permit, so the discount cannot change the decision within the allowed range. City Spin is cheaper at 4 hours and at 5 hours, discount or no discount. The 10% is a percent decrease applied to the total bill, not to the hourly rate alone.