calculus
posted by HELLO on .
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?

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(1180000/x^2)
dC/dx=0 when x^2 = 180000
x = 424
since C" > 0, it's a minimum. 
a) C=1000h+20x+25xh
b) h=3600/x
c) C=1000(3600/x)+20x+25x(3600/x)
C=3600000/x+20x+90000
dc/dx=3600000x^2+20
3600000x^2+20=0
3600000x^2=20
20x^2=3600000
x^2=180000
x=180000^1/2 (approx. 424.26)
to justify it do d2c/dx^2
d2y/dx^2=7200000x^3
sub in your answer
7200000*(180000^1/2)^3= 0.094
which is positive, therefore the point is a minimum. 
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 =