if you have 200 feet of fencing to enclose four adjacent rectangular enclosures. Determine what demensions should be used so that the enclosed area will be maximized.
adjacent how? in a single large rectangle, or an L-shape, or a long line?
single large rectangle steve
To determine the dimensions that will maximize the enclosed area, we can follow these steps:
1. Let's denote the width of each rectangular enclosure as "w".
2. Since there are four adjacent rectangular enclosures, there are three connections that require fencing.
- To enclose the two end enclosures, we need fencing for their left and right sides, totaling 2w.
- To enclose the two middle enclosures, we need the fencing for their top and bottom sides, totaling 2(w+2w)=6w.
- In total, we need 2w + 6w = 8w of fencing.
3. We are given that we have 200 feet of fencing. Setting up an equation based on the total length of fencing:
- 8w = 200
4. Solve the equation for "w":
- w = 200 / 8
- w = 25
5. Now, we have the width of each rectangular enclosure, which is 25 feet. To find the length of each enclosure, we need to consider the remaining 2w of fencing.
- Since there are two end enclosures, the total length of 2w is divided equally between them.
- Therefore, the length of each end enclosure is w = 25 feet.
6. The length of the two middle enclosures will be the same as their width plus the width of one end enclosure.
- Therefore, the length of each middle enclosure is (w + w + w) = (25 + 25 + 25) = 75 feet.
7. To summarize, the dimensions of each rectangular enclosure should be as follows:
- Width: 25 feet
- Length of end enclosures: 25 feet
- Length of middle enclosures: 75 feet
By using these dimensions, the enclosed area will be maximized.