pre cal help

Question

a gardener has 120 ft of deer resistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area enclosed to be at least 800 ft² what range of values is possible for the length of her garden?

Answer 1

The garden is to be rectangular shape.

 

Let x = width

Let y = length

 

 

Set the equations for perimeter and area.

 

2(x + y) = 120  -----> perimeter

 

x + y = 60   ------> perimeter

xy = 800   ---->  area

 

 

We have two equations to work with.

 

x + y = 60                eq1

xy = 800                   eq2

 

 

Substitute eq1 into eq2.  We can get eq2 in terms of x.

 

x(60 - x) = 800

 

-x² + 60x = 800

 

-x² + 60x - 800 = 0

 

 

Divide both sides of the equation by -1, so that we have a positive x².  The goal is to try to use FOIL to factor.

 

x² - 60x + 800 = 0

 

(x - 40)(x - 20) = 0

 

x = 40         and           x = 20

 

 

Next, we plug in these values of x into eq1 to solve for y.

 

y = 60 - x                     and                y = 60 - x

y = 60 - 40                                        y = 60 - 20

y = 20                                               y = 40

 

 

Based on this, the range of values for the length is

 

20 ≤ y ≤ 40