Suzanne has \(100\) m of electric fence to put up for a temporary pen for her horses. The location she is planning on using is along the side of the barn, so she will only need to use the fencing on three sides of the pen.

  1. Draw a diagram of the fence. Label the sides and identify the variables used.

    The perimeter of the fencing is \(100 \thinspace \rm{m}\). Because there are three sides, there will be two widths and only one length. Suppose \(w\) represents the width and \(l\) represents the length, then

    \(\begin{align}
    2w + l &= 100 \\ 
    l &= 100 - 2w \end{align}\)

    As a diagram, the fence would look like the image on the right.

  2. Write a quadratic function that represents the area of the pen.

    Let \(A\) represent the area of the pen, then \(\begin{array}{l}
    A = lw \\ 
    A = \left( {100 - 2w} \right)w \\ 
    A = 100w - 2w^2  \\ 
    A = -2w^2 + 100w \end{array}\)




  1. Sketch the graph of the area function.

    Using your graphing calculator, sketch the graph.

  2. What does the vertex represent?

    The vertex represents the value of w that will give the largest area, \(A\). In this case, the vertex is at \((25, 1 \thinspace 250)\), meaning that if the width is \(25\thinspace \rm{m}\), the length will be \(100 - 2(25) = 50 \thinspace \rm{m}\), and the area will be \(1\thinspace 250 \rm{m}^2\).

  3. What do the \(x\)-intercepts represent?

    The \(x\)-intercepts represent the width, \(w\), that will result in the area being \(0 \thinspace \rm{m}^2\). The two sets of dimensions are \(100 \thinspace \rm{m}\) by \(0 \thinspace \rm{m}\) and \(0 \thinspace \rm{m}\) by \(50 \thinspace \rm{m}\).

  4. What does the \(y\)-intercept represent?

    The \(y\)-intercept is \(0 \thinspace \rm{m}^2\).  When the width is \(0 \thinspace \rm{m}\), the area will be \(0 \thinspace \rm{m}^2\).

  5. What are the domain and range of the function?

    The domain of the function is limited because negative widths do not make sense and negative areas do not make sense. The domain is {\(w | 0 \le w \le 50, w \in\) R}.

    The range hits a maximum of \(1 \thinspace 250\) and a minimum of \(0\), therefore the range is {\(A | 0 \le A \le 1 \thinspace 250, A \in \) R}.

  6. Does the function have a maximum or minimum? What does it represent?

    This function has a maximum value of \(1 \thinspace 250\). It represents the maximum area that can be made using the \(100 \thinspace \rm{m}\) of fencing.