A security fence encloses a rectangular are on one side of a park in a city.

Question

A security fence encloses a rectangular are on one side of a park in a city. Three sides of fencing are used, since the fourth side of the area is formed by a building. The enclosed area measures 578 square feet. Exactly 68 feet of fencing is used to fence in three sides of this rectangle. What are the possible dimensions that could have been used to construct this area?

Answer 1

Call one side of the fencing x and call the other side y. Let's say that the rectangle's three sides are two x's and one y. Then:

 

Area = xy

578 = xy

 

Perimeter = 2x + y

68 = 2x + y

 

 

 

The perimeter equation tells us that y = 68 - 2x. Plug that into the area equation:

 

578 = x(68 - 2x)

578 = 68x - 2x²

2x² - 68x + 578 = 0

x² - 34x + 289 = 0

(x - 17)(x - 17) = 0

 

So, x = 17. That means y = 68 - 2(17) = 68 - 34 = 34. The dimensions are 17 × 34, with the 17-foot side doubled.