Question

Please help me with this problem I am not able to help my son to understand we keep getting it wrong please help.A farmer will build a rectangular pen for some goats. A wall will form one side of the pen. The farmer has 36 m of fencing to form the other three sides.The farmer plans to build the pen so that it has its maximum possible area. What will be the dimensions of the farmer’s goat pen?Enter your answer by filling in the boxes.

Answer 1

see the figure below to better understand the problem

The perimeter of the fence is equal to

2y+x=36 ------> x=36-2y -----> equation 1

The area is equal to

A=x*y -----> equation 2

substitute equation 1 in equation 2

A=(36-2y)*y

A=36y-2y^2 ----> quadratic equation

this quadratic equation represents a vertical parabola open downward

The vertex is a maximum

The y-coordinate of the vertex represents the maximum area

The x-coordinate of the vertex represents the dimension y of the pen for the maximum area

so

Find out the vertex of the parabola

Complete the square

`\begin{gathered} A=36y-2y^2 \\ A=-2y^2+36 \\ A=-2(y^2-18) \\ A=-2(y^2-18+9^2-9^2) \\ A=-2(y^2-18+9^2)+162 \\ A=-2(y-9)^2+162 \end{gathered}`

The vertex of the parabola is (9,162)

so

y=9

Find out the value of x

equation 1

x=36-2y

x=36-2(9)

x=18

the dimensions are 9 m x 18 m

Verify the maximum area

A=9*18=162 m2 ----> is ok is equal to the y-coordinate of the vertex

the answer is 9 m x 18 m