Question 965885
let l = the length and let w equal the width.


the area is equal to l*w.


since the area has to be 500 square feet, then you get 500 = l * w.


solve for w to get w = 500 / l.


the perimeter will be equal to l + 2w.


normally the perimeter is equal to 2l + 2w, but one of the lengths will be the side of the school, so you only need one l.


since w = 500 / l, the perimeter becomes equal to l + 2 * 500/l


set y = the perimeter and set x equal the length and the formula for perimeter becomes:


y = x + 2*500 / x


this can be simplified to y = x + 1000 / x


you want to minimize the perimeter which means you want to minimize y.


if you graph y = x + 1000 / x, you can see where y will be at a minimum.


the graph looks like this:


{{{graph(400,400,-150,150,-200,200,x + 1000/x)}}}


since both x and y have to be greater than 0, you are looking at the right side of the y-axis and above the x-axis on the graph.


your problem is to find the minimum point on the graph.


some graphing software will tell you what it is.


other graphing software will not.


in the absence of graphing software that will tell you, or a minimum point formula you can use, you need to use calculus to find it.


the derivative of the equation set to 0 will give you the min/max point of the equation.


the equation is y = x + 1000 / x


the derivative with respect to x is y' = 1 - 1000 / x^2


set this equal to 0 and you get 1 - 1000 / x^2 = 0


add 1000 / x^2 to both sides of this equation to get 1 = 1000 / x^2


solve for x^2 to get x^2 = 1000


solve for x to get x = plus or minus 31.6227766.


since x has to be positive, then x = 31.6227766.


since x represents the length, and the area is equal to the length * width, and the area is equal to 500, you get 500 = 31.6227766 * the width.


solve for the width to get the width = 15.8113883.


the perimeter is equal to the length plus 2 times the width, so the perimeter will be 31.6227766 + 2 * 15.8113883 which makes the perimeter equal to 63.2455532.


placing y = 63.2455532 on the grpah will show you that is the minimum point on the graph as shown below:


{{{graph(400,400,-150,150,-200,200,x + 1000/x,63.2455532)}}}


the least amount of fencing will be used when the length is equal to 31.62277766 and the width is equal to 15.8113883.


the area will be 500 square because 31.62277766 * 15.8113883 = 500.