A forest is burning in a valley. it is estimated that 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?

A. To calculate the cost of the fire, we need to consider both the cost of lost timber and the cost of firefighters. The cost of lost timber is given as $1000 per hour (h), so that part of the formula is 1000h.

The cost of firefighters includes transportation and salary/food. For each firefighter, the transportation cost is $20. The salary/food cost is $25 per hour (3600/x hours in this case). Therefore, the cost of firefighters is (20 + 25 * (3600/x)), and since there are x firefighters, we multiply this by x.

Combining the costs, the formula for C (cost of the fire) in terms of x and h is:
C(x, h) = 1000h + x(20 + 25 * (3600/x))

B. To find a formula that relates x and h, we need to consider the time it takes to stop the fire. It is given that x firefighters can stop the fire in (3600/x) hours.

For simplicity, let's define T as the time needed to stop the fire, then T = (3600/x). We can solve this equation for x by multiplying both sides by x, which gives us xT = 3600 or x = 3600/T.

So, the formula relating x and h is:
x = 3600/h

C. To minimize the cost C of the fire, we need to find the optimal number of firefighters x that minimizes the total cost. In our formula for C(x, h), the cost of lost timber increases linearly with h, and the cost of firefighters increases linearly with the number of firefighters.

As x increases, the cost of firefighters increases, and as h increases, the cost of lost timber increases. To find the minimum cost, we need to find a balance between these two costs.

To do this, we can differentiate the formula for C(x, h) with respect to x, set it to zero, and solve for x. The resulting x will give us the optimal number of firefighters that minimizes the cost C.

Taking the derivative of C(x, h) with respect to x, we have:
dC/dx = 20 + 25 * (3600/x) - 25 * (3600/x^2)

Setting dC/dx = 0, we can solve for x:
20 + 25 * (3600/x) - 25 * (3600/x^2) = 0

Simplifying this equation, we get:
20x^2 + 25 * 3600 - 90000 = 0

Solving this quadratic equation will give us the optimal value of x. Once we have x, we can substitute it back into the formula for C(x, h) to find the minimum cost C of the fire.