Uppose you need to minimize the cost of fencing in a rectangular region with a total area of 450 square feet. the material that will be used for three sides costs $15 per linear foot, and the material that will be used for the fourth side costs $27 per linear foot. write a function that expresses the cost of fencing the region in terms of the le

X = length of one of the 2 $30 sides y = length of one of the other sides ($30 or $15, doesn't matter)

Area = x * y = 500 ft2

Cost = (Length of Sides 1-3) * $30 + Length side 4 * $15

= (x + x + y) * $30 + y * $15

= (2x + y) * $30 + y*$15

= 60x + 30y + 15y

= 60x + 45y

Now we know Area = x * y = 500, so:

y = 500 / x

Substitute in 500/x for y:

Cost = C (x) = 60x + 45 * (500/x) = 60x + 22500/x

C (x) = 60x + 22500/x or C (x) = 60x + 22500*x^-1

This is your function!

Now take the derivative if you want to find x at the minimum cost. When the derivative is 0 you have reached a minimum in your cost:

dC/dx = 60 - 22500*x^-2 = 0

60 = 22500x^-2

x^-2 = 2.666 * 10 ^-3 Raise each side to the power of - 1/2 (X^-2) ^-1/2 = (2.666 * 10 ^-3) ^-1/2

X = 19.36 ft at the length of x at minimum cost. The minimum cost is about $2323.80