Question

Farmer Ed has 150 meters of fencing and wants to enclose a rectangular play that borders a river. If farmer Ed does not fence in the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Answer 1

150/3=50

50^2=2500m^2=largest area

Answer 2

Answer:

The largest area will be 2812.5 square meters.

Step-by-step explanation:

The perimeter of rectangle is given as:

$2L+2W$ (L is the length and W is the width)

As one side is not to be fenced, so the formula here will be : $P=L+2w$

Perimeter is 150.

So, $150=L+2W$ ; $L=150-2W$

Area of the rectangle is : $LW$

Plugging the value of L in the area formula;

Area = $(150 -2W)(W)$

This is a parabola or quadratic function whose maximum or minimum values occur at the average of the solutions.

So, Solving $(150 -2W)(W)=0$

=> $150 -2W = 0$ Or $W=0$

=> $150-2W=0$

=> $2W=150$

W = 75

So, the two solutions are zero and 75.

The average of them is $\frac{0+75}{2}=37.5$

Now, the maximum area is at W=37.5

And $L=150-2(37.5)$

L = 75

The dimensions that maximize the area are L=75 and width W=37.5

And maximum area = $75\times37.5$ = 2812.5 square meters

Hence, the largest area will be 2812.5 square meters.