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).

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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 =