A farmer wants to fence in three sides of a rectangular field shown below with 880 feet of fencing. The other side of the rectangle will be a river. If the enclosed area is to be maximum, find the dimensions of the field.
440x220 with the 440' side parallel to the river.
the related questions below provide steps for the solution.
Let's solve this step by step:
1. Let's assume the length of the rectangular field is L, and the width is W.
2. There are three sides of the rectangle that need to be fenced, so the total length of the three sides is L + W + L = 2L + W.
3. According to the given information, the total length of fencing available is 880 feet.
4. We can write the equation as follows: 2L + W = 880.
5. Now, we need to maximize the enclosed area, which is given by A = L * W.
6. To find the maximum area, we can use the equation in step 4 and isolate the variable W: W = 880 - 2L.
7. Substituting this value of W in the formula for area: A = L * (880 - 2L) = 880L - 2L^2.
8. To find the maximum value of A, we need to find the value of L that maximizes the area. We can do this by finding the vertex of the parabola formed by the equation in step 7.
9. The vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b/2a.
For our equation, the coefficient of L^2 is -2 and the coefficient of L is 880. So, the x-coordinate of the vertex is x = -880/(2*(-2)) = 220.
10. This value of L will give us the maximum area, but we also need to find the corresponding value of W. We can substitute the value of L in the equation from step 6 to find W: W = 880 - 2L = 880 - 2(220) = 440.
11. Therefore, the dimensions of the field that will maximize the enclosed area are Length = 220 feet, and Width = 440 feet.
To find the dimensions of the field that would maximize the enclosed area, we need to consider the geometry of the problem. Let's break it down step by step:
1. Define the variables:
- Let's represent the width of the field as "w" (in feet).
- Let's represent the length of the field as "l" (in feet). The length will be parallel to the river.
2. Identify the constraints:
- We have 880 feet of fencing available to enclose three sides of the field.
- Two sides of the field will be equal in length and will be equal to the width of the field (w).
- The other side of the field will be the length of the field (l).
3. Set up the equation for the perimeter:
- The total length of the three sides that need to be fenced is given as 880 feet.
- The equation for the perimeter is: Perimeter = 2w + l + w. Since two sides are of equal length, the equation can be simplified to: Perimeter = 2w + l.
4. Substitute the given information into the equation:
- We are given that the total length of the three sides to be fenced is 880 feet: 2w + l = 880.
5. Express one variable in terms of the other:
- Solve the equation for l: l = 880 - 2w.
6. Express the objective function:
- The area of a rectangle is given by the equation: Area = length x width. In this case, the area can be expressed as: Area = w(880 - 2w).
7. Maximize the objective function:
- To maximize the area, we need to find the value of w that makes the derivative of the area function equal to zero.
- Differentiate the area function using calculus: d(Area)/dw = 880 - 4w.
- Set the derivative equal to zero and solve for w: 880 - 4w = 0.
- Solve for w: w = 880/4 = 220.
8. Find the corresponding value of l:
- Substitute the value of w (220) into the equation for l: l = 880 - 2w = 880 - 2(220) = 440.
Thus, the dimensions of the field that would maximize the enclosed area are: width = 220 feet and length = 440 feet.