To express the area $A$ of the rectangular garden as a function of the width $x$, follow these steps:

1. Let $x$ be the width of the rectangle.
2. Let $y$ be the length of the rectangle.
3. The perimeter of the rectangle is the total length of the fencing, which is given as 200 yards. The formula for the perimeter $P$ of a rectangle is: $P = 2x + 2y$ Substituting $P = 200$: $2x + 2y = 200$
4. Solving for $y$ in terms of $x$: $2y = 200 - 2x$ $y = 100 - x$
5. The area $A$ of the rectangle is the product of its length and width: $A = x \cdot y$ Substituting $y = 100 - x$ into the area formula: $A(x) = x(100 - x)$ $A(x) = 100x - x^2$

Thus, the area $A$ as a function of the width $x$ is: $A(x) = 100x - x^2$

Do you have any questions, or would you like further details?

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Here are 5 related questions for practice:

1. What is the maximum area that can be enclosed by the 200 yards of fencing?
2. How do you find the width that maximizes the area?
3. What would the area function look like if the perimeter were 300 yards instead?
4. How would the problem change if the farmer wanted to divide the garden into two equal parts using an additional fence parallel to the width?
5. What happens to the area as the width $x$ approaches 0?

Tip: To maximize the area, you can find the vertex of the quadratic function $A(x) = 100x - x^2$.