A sports store sells about 50 mountain bikes per month at a fee of $220 each. For each $20 decrease in fee, about 10 more bikes per month are sold.
1) Write a quadratic function in standard form that models the revenue from bike sales?
2) What fee produces the maximum revenue?
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The revenue is projected to be
R = ( 50 + 10 n ) ( 220 – 20 n )
To find where R has a maximum, take the first derivative, set it equal to zero, and solve for n:
R = ( 50 + 10 n ) ( 220 – 20 n )
R = -200 n² + 1200 n + 11,000
dR/dn = -400 n + 1200 = 0
n = 3
It looks like the maximum revenue is achieved by selling 80 bikes at $160 each.
Reality check:
90 bikes at $140 yields $12,600
80 bikes at $160 yields $12,800
70 bikes at $180 yields $12,600
So it looks like the sweet spot is indeed at about $160 per bike with 80 bikes sold.