Fenced in garden (read in description for full question)

Question

Angela has 140 feet of fencing. She wants to enclose a rectangular vegetable garden and she wants the enclosed area to be at least 1000 square feet. What ranges of values are possible for the length of her garden?

Answer 1

We have a rectangular shape.  The length is the same as perimeter.  The area is a minimum of 1000 square feet.

 

 

Perimeter = 140 feet

 

Area = 1000 square feet

 

 

Let length = x

Let width = y

 

 

We can write formulas using the information.

 

2x + 2y = 140          perimeter

 

x + y = 70

 

xy = 1000               area

 

 

We have two equations to work with.

 

x + y = 70                eq1

xy = 1000                  eq2

 

 

Substitute eq1 into eq2.  Lets get eq2 in terms of x.

 

x(70 - x) = 1000

 

 

-x² + 70x = 1000

 

-x² + 70x - 1000 = 0

 

 

Use the quadratic formula to solve for x.

 

x = (-70 ± √(4900 - 4(1000))) / -2

 

x = (-70 ± √(4900 - 4000)) / -2

 

x = (-70 ± √(900)) / -2

 

x = (-70 ± 30) / -2

 

x = (-70 - 30) / -2               and              x = (-70 + 30) / -2

 

x = -100 / -2                       and               x = -40 / -2

 

x = 50                                 and              x = 20

 

 

Now we can substitute these values of x into eq2.

 

y = 1000 / 50                     and                y = 1000 / 20

y = 20                                                      y = 50

 

 

First pair of dimensions:

length = 50 feet

width = 20 feet

 

 

Second pair of dimensions:

length = 20 feet

width = 50 feet

 

As you can see, the lengths are between 20 and 50 feet included.

 

The range of values that are possible for the length are between 20 feet and 50 feet, including 20 feet and 50 feet.