On the side of his house, a man is going to build two enclosures. One enclosure for a garden and another enclosure to keep his dogs in so the dogs aren’t roaming around. It is illegal to have any dogs roaming around. He bought 180ft of fence to use and one side is already existing. Refer to the picture on the next page. Since there are three short sides and one long side, to set up the perimeter, the short sides will be x and the long side will be y. So the equation is 3x + y = 180.
The 180 comes from the 180 ft of fence the man already has. To find the total area, we need to solve the equation for y. So we subtract 3x from one side to the other. So the equation will be y = 180 – 3x The area formula is A = xy. So you substitute the final equation you got above with the y in the area formula to end up with A = x(180-3x). Then, you multiply everything in the parenthesis by x. You get 180x – 3x^2. To find the domain, you divide the total amount of material by the number of x’s. 80 is the total amount of material and there are 3 x’s. So divide 180 by 3. The answer is 60. So the domain is (0, 60). To find the dimensions that yield maximum area, on the calculator press the y= button and type in the final area equation we got, which is 180x – 3x^2. Then before you graph, you press the window button and enter the domain in the Xmin and Xmax slots. So it would look like Xmin = 0 and Xmax = 60. You just leave the other ones alone.
Then you press zoom and 0 and it shows the graph. Refer to graph on other page. Then you press the trace button and you get the dimensions x= 30 and y= 2700. Those are your dimensions for maximum area. To find the maximum area, all you have to do is look at your dimensions. The calculator gives you x= 30 and y= 2700. The maximum area is the y you are given, which is 2700. So your maximum area is 2,700 sq. yds. x= 30 yds and y= 90yds. The picture that represents the situation: The graph: