### Instruction: Richard is building a rectangular backyard from 360 feet of fencing. The fencing must cover three sides of the backyard (the fourth side is bordered by Richard's house). What is the maximum area of this backyard?### Response: I want to maximize the area of the backyard, which is length times width.
I also have a constraint on the amount of fencing I can use, which is 360 feet.
If I let x be the length of the backyard and y be the width, then the area is A = xy and the fencing constraint is 2x + y = 360.
I can use the constraint to eliminate one variable and express the area in terms of the other.
For example, I can solve for y and get y = 360 - 2x, and then plug that into the area formula to get A = x(360 - 2x) = 360x - 2x^2.
Now I have a quadratic function for the area in terms of x, and I want to find its maximum value.
I know that a quadratic function has a maximum (or minimum) when x is equal to the opposite of the coefficient of x divided by twice the coefficient of x^2.
This is called the vertex formula.
In this case, the coefficient of x is 360 and the coefficient of x^2 is -2, so the vertex formula gives me x = -360 / (2 * -2) = 90.
This means that the maximum area occurs when x = 90, and I can plug that back into the area formula to find the corresponding value of A. A = 90(360 - 2 * 90) = 90 * 180 = 16,200.
I also need to find the value of y when x = 90, which I can do by using the fencing constraint.
y = 360 - 2 * 90 = 180.