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?

Let's break this down step by step:

1. **Picture Description**: You want to enclose a rectangular field using 16 miles of fence. Suppose you arrange it into a rectangle with a width of 3 miles and a length of 5 miles.

   - The total perimeter \( P \) of the rectangle is the sum of all sides:
   \[
   P = 2(\text{Length}) + 2(\text{Width})
   \]
   - Substituting the given values:
   \[
   P = 2(5) + 2(3) = 10 + 6 = 16 \text{ miles}
   \]
   This confirms that the fence can indeed form a rectangle of length 5 miles and width 3 miles.

2. **Area Calculation**: The area \( A \) of the rectangle is given by:
   \[
   A = \text{Length} \times \text{Width} = 5 \times 3 = 15 \text{ square miles}
   \]

So, the area of the rectangle is **15 square miles**.

Would you like a more detailed explanation or a visual diagram of this setup?

### Related Questions:
1. What if the width is 4 miles? What would the corresponding length be?
2. How does the area change if you increase the width to 4 miles while keeping the perimeter constant?
3. What is the maximum area you can enclose with 16 miles of fencing?
4. Can you express the area as a function of one variable, say the width?
5. How would the formula for the perimeter change if it were an irregular quadrilateral?

**Tip**: For a fixed perimeter, a square encloses the maximum area among rectangles.

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

Explore how to enclose a rectangular field with 16 miles of fence, arranging it into a 5-mile length and 3-mile width. Learn to calculate the perimeter and area, and understand how different dimensions affect the total enclosed space. Perfect for students in Grades 6-8.

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