Question

A farmer has 1,800 feet of fencing and wants to fence off a rectangular field that borders a straight river. he needs no fence along the river. write the function that will produce the largest area if x is the short side of the rectangle.

Answer 1

-2x^2 + 1800x

Explanation:

2x + l = 1800

l = 1800 - 2x

l * x = area

x(1800 - 2x) = area

-2x^2 + 1800x

Answer 2

There are 2 short sides x and 1 long side y.

x+x+y=1,800
2x+y=1,800
y=1,800-2x

The area of the rectangle is A=x(1,800-x).


A is a function with variable x. Moreover it is a quadratic (second degree) function.

The graph of the function is a parabola which opens downwards, because the signs of the leading coefficients of the factors, multiply to minus:

(x)(-x)=-x^2.


This means that the highest point of the parabola, is its vertex V(h,k), so the function takes its largest value for x=h.


The roots of A=x(1,800-x) are x=0 and x=1,800, thus the x coordinate of the vertex, must be the midpoint of 0 and 1,800, that is 900.


Answer:

A=x(1,800-x)

largest area is when x=900