posted by km on .
Mike's family wants to build a rectangular fenced backyard area for their dog. They have a 20-meter length of wire fence and four posts. They can also use the 20-meter straight length of the back of their house as a side of the enclosure, but the fence cannot attach directly to the house. The fence must stretch taught between posts, and they have fasteners to attach the fence to the posts. Describe or sketch a design for the dog enclosure that yields the maximum area using the resources they have. List the dimensions of the enclosure and list the total area. Check your work by describing a similar design that does not enclose as much area.
the conditions require that the 4 posts are the corners of the rectangle. So, if the width is x, and the length is along the house, the fence used is x+x+(20-2x)
area = x(20-2x) = 2x(10-x)
the max is achieved when x is midway between the roots, or x=5.
The pen is thus 5x10, with area = 50
suppose the width is 6. The pen is then 6x8 with area = 48
suppose the width is 4. Then the pen is 4x12 with area = 48.