SOLUTION: A health club currently charges its 2,000 clients monthly membership dues of $45. The board of directors decides to increase the monthly membership dues. Market research shows that

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Question 906178: A health club currently charges its 2,000 clients monthly membership dues of $45. The board of directors decides to increase the monthly membership dues. Market research shows that each $1 increase in dues will result in the loss of 7 clients. How much should the club charge each month to optimize the revenue from monthly dues?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A health club currently charges its 2,000 clients monthly membership dues of $45.
The board of directors decides to increase the monthly membership dues.
Market research shows that each $1 increase in dues will result in the loss of 7 clients.
How much should the club charge each month to optimize the revenue from monthly dues?
:
let x = no. or dollar increases
and
7x = no. clients lost
:
Rev = f(x)
f(x) = (45 + x) * (2000-7x)
FOIL
f(x) = 90000 - 315x + 2000x - 7x^2
f(x) = 90000 + 1685x - 7x^2
As quadratic equation where a = -7 and b = 1685 the axis of symmetry will give the max revenue x = -b/(2a)
x = %281685%29%2F%282%2A-7%29
x = +120.357 ~ 120
:
Max revenue when: 120 + 45 = $165 a month with
2000 - 7(120) = 842 clients
Rev then: 842 * 165 = $138,930