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how would i do this quadratic word problem

A local daycare centre charges $65 per day to care for a child. The daycare currently cares for 150 children per day. A survey shows that the enrollment in the daycare will increase by 10 children for each $5 decrease in daily fee, and would decrease similarly if the daily fees are increased.

What daily fee would result in the greatest revenue for the daycare centre?

  • math -

    Let the number of $5 decreases be n
    so cost per child = 65 - 5n
    number of children = 150 + 10n

    Revenue = (65-5n)(150+10n)
    = 9750 - 100n - 50n^2
    d(Revenue)/dx = -100 -100n = 0 for a max/min of Revenue

    100n = -100
    n = -1
    so the number of decreases is -1, or there should be an increase of
    $5 for a daily fee of $70

    check:
    at $65 , number is 150, R = 9750
    at $60, number is 160, R = 9600
    at $70, number is 140, R = 9800
    at $75, number is 130, R = 9750

  • math -

    x = amount of increase of fee
    enrollment = n = 150 - 10 (x/5)
    = 150 - 2x
    revenue = n * (65+x)
    so
    revenue = (150-2x)(65+x)
    R = 9750 + 20x -2 x^2
    find the vertex
    R/2 = 4875 +10 x -x^2
    x^2 -10 x = -(R/2) + 4875
    x^2 - 10 x + 100 = -(R/2) +4975
    (x-10)^2 =
    so vertex at x = 10
    so fee at vertex = 65+10 = 75

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