A farmer wants to enclose the largest possible rectangle using a straight river as one boundary and #2400# feet of fencing. What are the dimensions of the rectangle of largest possible area?
The maximum area is given by a
Method 1 - Geometry
If there were no river then the maximum area would be given by a square configuration since increasing the length and decreasing the width of a square by the same value
Now add the river on one side. The river acts like a mirror. As we fence in a rectangle on one side of the river, imagine an identical rectangle on the other side of the river. The maximum area is given when the two rectangles form a square, that is when we have a rectangle twice as long as wide.
Method 2 - Algebra and calculus
Suppose the long side of the rectangle has length
#a(t) = (2400 - t)/2 * t = -1/2t^2 + 1200t#
Since the coefficient of the leading
So the longest side is
The dimensions is 600 ft by 1200 ft.
First, set up the equations:
perimeter= 2x+ y =2400 (remember, the river serves as a natural border so the fence only has 3 sides with two of them being equivalent because they are opposite sides of a rectangle.)
Use substitution to get one variable.
2x+ y= 2400
y= 2400- 2x
Plug that into the area equation: xy
distribute the x
There are two ways to do this problem: using calculus or using the calculator.
Using calculus, to find the maximum of the area, the derivative of the area equation should equal zero.
area = 2400x- 2
Set it to zero and solve.
2400 -4x = 0
solve for y by plugging x into the perimeter equation
2(600) +y = 2400
1200 + y= 2400
Using the calculator, push the Y= button and enter the area equation, 2400x- 2
You might have to adjust the window to see the full parabolic curve. To calculate the maximum of the area, press the 2ND and then the TRACE button. Then hit number 4 to calculate maximum. It will ask you for left bound which means go to the left of the maximum of the curve and hit enter. Then it will ask you right bound, which means to the left of the maximum and hit enter. It will then ask GUESS? which means go in-between the two left and right bound and hit enter.
It will yield (600, 720000) which means x= 600. Plug that into the perimeter equation to get a y value of 1200. Those are the dimensions of the fence to maximize area.