a rectangular pen with partitions is to be built from 800 feet of fencing. Express the area as a function of x alone. graph over a reasonable domain and find the dimensions of the pen of the maximum area that an be built under these conditions

how many partitions? Is the fence used to separate the partitions? And what do you want x to represent?

three partitions and x is shown as dimension of a partition

To solve this problem, let's break it down step by step:

1. Understanding the problem:
- We need to build a rectangular pen with partitions.
- We have 800 feet of fencing available.
- We want to find the dimensions of the pen that maximize its area.

2. Formulating the problem:
- Let's denote the length of the rectangular pen as L and the width as W.
- The perimeter of the pen will be equal to the total amount of fencing available: 2L + 3W = 800. (Explanation: Each length will have two sides, and each width will have three sides because of the partitions.)
- We need to express the area of the pen, A, in terms of x alone.

3. Expressing the area as a function of x:
- Since we have two variables (L and W), we need to eliminate one of them.
- Solving the equation 2L + 3W = 800 for L gives us L = (800 - 3W) / 2.
- The area of the pen is given by A = L * W.

Substituting L with (800 - 3W) / 2, we get:
A = (800 - 3W) / 2 * W.

Simplifying further, we can express the area of the pen as a function of x alone:
A = 400W - (3/2)W^2.

4. Graphing the function:
- To graph the function A = 400W - (3/2)W^2, we can use a graphing calculator or software.
- Choose a reasonable domain, such as 0 ≤ W ≤ 400, since the width cannot be negative and should not exceed 400 feet (to ensure that we have enough fencing).

5. Finding the dimensions of the pen with maximum area:
- To find the dimensions of the pen that maximize its area, we need to locate the highest point on the graph.
- Locate the vertex of the parabolic function A = 400W - (3/2)W^2, which will give us the values of W and A at the maximum point.

6. Conclusion:
Based on the graph, the dimensions of the rectangular pen with partitions that maximize the area can be found by:
- Locating the vertex (W, A) on the graph.
- The width, W, represents one of the dimensions of the pen.
- You can find the corresponding length, L, using the equation L = (800 - 3W) / 2.