18) a rectangular dog pen is constructed using a barn wall as one side and 60m of fencing for the other…

Explanation:

Step1: Set up variables

Let the side parallel to the barn wall be (x) meters and the other two equal - length sides be (y) meters. Then the fencing equation is (x + 2y=60), so (x = 60 - 2y). The area formula (A=xy). Substitute (x) into the area formula: (A=(60 - 2y)y=60y-2y^{2}).

Step2: Find the maximum of the quadratic function

For a quadratic function (A(y)=ay^{2}+by + c) (here (a=-2), (b = 60), (c = 0)), the vertex of the parabola (y=-\frac{b}{2a}). Substitute (a=-2) and (b = 60) into (y =-\frac{b}{2a}), we get (y=-\frac{60}{2\times(-2)} = 15).

Step3: Find the value of (x)

Substitute (y = 15) into (x=60 - 2y), then (x=60-2\times15=30).

Step4: Calculate the area

Substitute (x = 30) and (y = 15) into (A=xy), (A=30\times15 = 450).

Answer:

The dimensions of the pen are (30m) (parallel to the barn wall) and (15m) (perpendicular to the barn wall), and the maximum area is (450m^{2}).