Let's break down the problem step by step.

Given Information:

• The farmer has 150 yards of fencing to use.
• The garden is rectangular.
• There's an opening that is $\frac{1}{3}$ of the garden's length.

Task:

1. Write a function $A(x)$ that describes the area of the garden, where $x$ is the length of the garden.
2. Find the dimensions that maximize the area and calculate the maximum area.

Step 1: Define Variables

• Let $x$ be the length of the garden.
• Let $y$ be the width of the garden.

Step 2: Total Fence Length

The total perimeter, accounting for the opening, would be:

• Length for fencing on the length side: $x - \frac{x}{3} = \frac{2x}{3}$
• Fencing used for the width: Two widths $= 2y$

The total amount of fencing available is given as 150 yards, so: $\frac{2x}{3} + 2y = 150$

Step 3: Solve for Width (y)

Solve this equation for $y$: $2y = 150 - \frac{2x}{3}$ $y = \frac{150 - \frac{2x}{3}}{2}$ Simplify: $y = 75 - \frac{x}{3}$

Step 4: Area Function

The area of the garden is given by the formula for the area of a rectangle: $A(x) = x \cdot y$ Substitute $y = 75 - \frac{x}{3}$ into the area formula: $A(x) = x \left( 75 - \frac{x}{3} \right)$ Simplify: $A(x) = 75x - \frac{x^2}{3}$

This is the function that describes the area of the garden in terms of $x$.

Step 5: Maximize the Area

To maximize the area, we need to find the critical points of $A(x)$ by differentiating it with respect to $x$ and setting the derivative equal to 0.

1. Differentiate: $A'(x) = 75 - \frac{2x}{3}$
2. Set $A'(x) = 0$: $75 - \frac{2x}{3} = 0$
3. Solve for $x$: $75 = \frac{2x}{3}$ $x = 112.5 \, \text{yards}$

Step 6: Find Width (y) and Maximum Area

Now, find the width by substituting $x = 112.5$ into the equation for $y$: $y = 75 - \frac{112.5}{3} = 75 - 37.5 = 37.5 \, \text{yards}$

The dimensions of the garden that maximize the area are:

• Length = 112.5 yards
• Width = 37.5 yards

The maximum area is: $A(112.5) = 112.5 \times 37.5 = 4218.75 \, \text{square yards}$

Final Answer:

• Length = 112.5 yards
• Width = 37.5 yards
• Maximum Area = 4218.75 square yards

Would you like further details on any step or clarification?

5 Related Questions:

1. How does the derivative help in finding the maximum area?
2. Could the length be less than 112.5 yards and still result in maximum area?
3. How does the opening size (1/3 of the garden’s length) affect the overall area?
4. How would the solution change if the total fencing length were different?
5. Can the width ever exceed the length in this configuration?

Tip: Always ensure that the function is correctly simplified before differentiating when optimizing geometrical problems like this!