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A forest is burning in a valley. If the fire burns for h hours, the cost of the lost timber is 1000h dollars. They estimate that x firefighters can stop the fire in 3600/x hours. The cost for each firefighter is $20 (transportation)plus $25 per hour (salary/ food).
A. Let C be the cost of the fire. C will depend on both h and x . Give a formula for C in terms of x and h.
B. Give a formula that relates x and h.
C. How may firefighters should be used if the cost C of the fire is to be minimized?

  • calculus -

    so, given x firefighters,

    h = 3600/x
    C(h,x) = 1000h + 20x + 25hx
    C(x) = 1000(3600/x) + 20x + 25(3600/x)x
    = 3600000/x + 20x + 25*3600

    dC/dx = -3600000/x^2 + 20
    = 20(1-180000/x^2)
    dC/dx=0 when x^2 = 180000
    x = 424

    since C" > 0, it's a minimum.

  • calculus -

    a) C=1000h+20x+25xh

    b) h=3600/x

    c) C=1000(3600/x)+20x+25x(3600/x)
    x=180000^1/2 (approx. 424.26)

    to justify it do d2c/dx^2


    sub in your answer

    7200000*(180000^1/2)^-3= 0.094

    which is positive, therefore the point is a minimum.

  • calculus -

    A. The cost of the timber lost = 1000 h
    The cost to have x firemen is x(25 h +20)
    The time to stop the fire is t= 3600/x

    C= 1000 h + x(25h + 20)

    B. Set h = t

    H = 3600/x

    C. Substitute for x in cost equation

    C= 1000 h + x(25h + 20) = 1000*3600/x + x(25*3600/x +20) = 3.6*10^6/x + 25*3600 +20x

    Take dC/dx and set the result equal to zero

    DC/dx = 0= -3.6*10^6/x^2 +20. ----> x =sqrt(1.8*10^5) = 424
    C =

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