A farmer decides to enclose a rectangular garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 100 ft of fence? What should the dimensions of the garden be to this given area?
The maximum area that can be enclosed with 100 ft?
The dimensions of the garden to give this area is 50ft by ?
Yup
To find the maximum area that can be enclosed with 100 ft of fence, we need to determine the dimensions of the rectangular garden that will give us the largest possible area. Let's break down the problem step by step:
1. Consider the given scenario: The farmer wants to enclose a rectangular garden, using the side of a barn as one side of the rectangle. This means that 3 sides of the rectangle will be enclosed by the fence, and the fourth side will be the barn itself.
2. Let's denote the length and width of the rectangular garden as L and W, respectively.
3. Since the side of the barn is already included as one side of the rectangle, we only need to fence the other three sides (two widths and one length). This means that the total length of fence used will be: 2W (for the widths) + L (for the length) = 2W + L.
4. According to the problem, the total length of fence available is 100 ft. Therefore, we can write the equation: 2W + L = 100.
5. Now, let's express the area of the rectangular garden in terms of L and W. The area (A) of a rectangle is calculated by multiplying the length (L) by the width (W), so A = L * W.
6. We want to find the maximum possible area, so we need to maximize the expression A = L * W.
7. To solve for the maximum area, we can use the equation from step 4 to express one variable in terms of the other. For example, we can solve for L in terms of W: L = 100 - 2W.
8. Substituting this expression for L into the area equation, we get: A = (100 - 2W) * W.
9. Now, we have an equation for the area in terms of only one variable, W. To find the maximum area, we can plot this equation as a quadratic function and determine the value of W that yields the vertex of the parabola.
10. The vertex of the parabola corresponds to the maximum area. In this case, the quadratic equation is in the form ax^2 + bx + c. Comparing it to our equation A = (100 - 2W) * W, we see that a = -2, b = 100, and c = 0.
11. The x-coordinate of the vertex is given by the formula x = -b / (2a). In our case, x represents W, so we have W = -b / (2a) = -100 / (2 * -2) = 25.
12. The corresponding value of L can be found by substituting W = 25 back into the equation from step 4: L = 100 - 2W = 100 - 2 * 25 = 100 - 50 = 50.
Therefore, the maximum area that the farmer can enclose with 100 ft of fence is 1250 sq. ft. and the dimensions of the garden to achieve this area are 50ft by 25ft.