Homework Help: calculus

Posted by HELLO on Friday, January 4, 2013 at 7:21pm.

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 - Steve, Saturday, January 5, 2013 at 3:35pm
so, given x firefighters,

h = 3600/x
C(h,x) = 1000h + 20x + 25hx
so,
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.

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