Investigation 3.3.1. Perimeter and Area.
Do all rectangles with the same perimeter, say 36 inches, have the same area? Two different rectangles with perimeter 36 inches are shown. The first rectangle has base 10 inches and height 8 inches, and its area is 80 square inches. The second rectangle has base 12 inches and height 6 inches. Its area is 72 square inches.

The table shows the bases of various rectangles, in inches. Each rectangle has a perimeter of 36 inches. Fill in the height and the area of each rectangle. (To find the height of the rectangle, reason as follows: The base plus the height makes up half of the rectangle’s perimeter.)
Base Height Area \(10\) \(8\) \(80\) \(12\) \(6\) \(72\) \(3\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(14\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(5\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(17\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(19\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(2\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(11\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(4\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(16\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(15\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(1\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(6\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(8\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(13\) \(\hphantom{0000}\) \(\hphantom{0000}\) \(7\) \(\hphantom{0000}\) \(\hphantom{0000}\)  What happens to the area of the rectangle when we change its base? On the grid above, plot the points with coordinates (Base, Area). (For this graph we will not use the heights of the rectangles.) The first two points, \((10, 80)\) and \((12, 72)\text{,}\) are shown. Connect your data points with a smooth curve.
 What are the coordinates of the highest point on your graph?
 Each point on your graph represents a particular rectangle with perimeter 36 inches. The first coordinate of the point gives the base of the rectangle, and the second coordinate gives the area of the rectangle. What is the largest area you found among rectangles with perimeter 36 inches? What is the base for that rectangle? What is its height?
 Describe the rectangle that corresponds to the point \((13, 65)\text{.}\)
 Find two points on your graph with vertical coordinate 80.
 If the rectangle has area 80 square inches, what is its base? Why are there two different answers here? Describe the rectangle corresponding to each answer.
 Now we’ll write an algebraic expression for the area of the rectangle in terms of its base. Let \(x\) represent the base of the rectangle. First, express the height of the rectangle in terms of \(x\text{.}\) (Hint: If the perimeter of the rectangle is 36 inches, what is the sum of the base and the height?) Now write an expression for the area of the rectangle in terms of \(x\text{.}\)
 Use your formula from part (8) to compute the area of the rectangle when the base is 5 inches. Does your answer agree with the values in your table and the point on your graph?
 Use your formula to compute the area of the rectangle when \(x=0\) and when \(x=18\text{.}\) Describe the “rectangles” that correspond to these data points.
 Continue your graph to include the points corresponding to \(x=0\) and to \(x=18\text{.}\)