Enclosing a field: You have 16 miles of fence that you will use to enclose a rectangular field. a. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 3 miles...

Enclosing a field: You have 16 miles of fence that you will use to enclose a rectangular field.

a. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 3 miles and length 5 miles. What is the area of this rectangle?

b. Draw a picture to show that you can arrange the 16 miles of fence into a rectangle of width 2 miles and length 6 miles. What is the area of this rectangle?

c. The first two parts of this exercise are designed to show you that you can get different areas for rectangles of the same perimeter, 16 miles. In general, if you arrange the 16 miles of fence into a rectangle of width w miles, then it will enclose an area of A = w(8 − w) square miles.

 i. Make a graph of area enclosed as a function of w, and explain what the graph is showing.

 ii. What width w should you use to enclose the most area?

iii. What is the length of the maximum-area rectangle that you have made, and what kind of figure do you have?