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 find the cost of the fire, we need to consider both the cost of the lost timber and the cost of the firefighters. Let's first calculate the cost of the lost timber. It is given as 1000h dollars for every hour the fire burns. So, the cost of the lost timber is simply 1000h.

Now, let's calculate the cost of the firefighters. For each firefighter, there is a transportation cost of $20 plus a salary/food cost of $25 per hour. Since there are x firefighters, the total cost for the firefighters can be calculated as follows: x * ($20 + $25 * (3600/x)).

To find the total cost, C, we need to sum up the cost of the lost timber and the cost of the firefighters: C = 1000h + x * ($20 + $25 * (3600/x)).

B. To relate the number of firefighters, x, and the time it takes to stop the fire, h, we can use the information given in the question. It states that x firefighters can stop the fire in 3600/x hours. This gives us the equation: h = 3600/x.

C. To minimize the cost, C, we need to determine the number of firefighters that will result in the minimum cost. To do this, we can take the derivative of C with respect to x and set it equal to zero.

dC/dx = derivative of (1000h + x * ($20 + $25 * (3600/x))) with respect to x.

By simplifying the expression and taking the derivative, we get:

dC/dx = -25 * 3600/x^2 + $20.

Setting dC/dx equal to zero, we have:

-25 * 3600/x^2 + $20 = 0.

Simplifying further, we get:

-25 * 3600 = $20 * x^2.

Solving for x^2, we have:

x^2 = (25 * 3600) / $20.

x^2 = 4500.

Taking the square root of both sides, we find:

x = √4500.

Therefore, the number of firefighters that should be used to minimize the cost C of the fire is approximately x = 67 (rounded to the nearest whole number).