You have a 1200 ft roll of fencing and a large field. You want to make a rectangular area split in half to contain two padlocks.

Question

Solve a quadratic equation to determine the dimensions of the largest enclosed area.

I need the steps for this question. 

Answer 1

You have two equations of interest here: perimeter and area. First of all, you are making a rectangular area with fencing, so you know the perimeter of your field must be equal to the amount of fencing you have. So, if you let x be the length and y be the width of your rectangle, then you know that

 

 (1)               2x+2y=1200.

 

Also, you have an area equation:

 

 (2)               A(x,y)= xy     (area equals length times width)

 

We want to maximize the second equation. First, solving equation (1) for y gives us:

 

 y=600-x

 

Plugging this value into (2) yields:

 

 A(x)=x(600-x)=600x-x^2. This is the quadratic equation that you must "solve"

 

In order to obtain the dimensions for largest area, you must maximize this new area equation. Two methods that come to mind are:

 

 --Take derivative and set equal to 0 and solve for x            OR

 

--use vertex formula (since this quadratic equation has a maximum at its vertex)

 

Using the first option and taking the derivative yields:

 

 A'(x)=600-2x.

 

Setting this equation equal to 0 gives us

 

 600-2x=0  which simplifies to x=300.

 

Since we knew that y=600-x, plugging in x=300 yields that y=300. So the dimensions in question are x=y=300.

 

Note that this gives us the area of the entire rectangular figure (Area=300*300=90000). Splitting the area in half gives us the area of each of the two pieces created.