what is the largest area that can be enclosed?

Question

you have 600 feet of fencing to enclose a rectangular plot that borders on a river. you are not going to fence the side along the river.

Answer 1

 

Let X = each of the 2 congruent sides

Let 600 - 2X = the 3rd side 

 

NOTE:  600 - 2X + X + X = 600 .... the total fencing

 

                         X

--------------------------------------------

R                                                    |

I                                                     |

V                                                    | 600 - 2x

E                                                    |

R                                                    |

--------------------------------------------

                         X

 

Area = Length • Width

        = X(600 - 2X)

        = -2X² + 600X

 

This is a quadratic function that opens downward (a is negative).  The maximum value occurs at the vertex

 

The x-coordinate of the vertex is -b/2a

 

      -b/(2a) = -600/(2• -2) = 600/4 = 150

 

So the maximum area occurs when 2 of the sides are 150 ... the third side is 300

 

The maximum area = 300•150 = 45,000

 

We can also find the maximum area by plugging 150 in for x in our constructed quadratic formula

 

         -2(150)² + 600(150)

     = -2(22500)+ 90,000

     = -45,000 + 90,000

     = 45,000