A farmer wants to enclose three sides of a rectangular area that borders a creek. He has 2400 meters of fencing material. What is the maximum area that can be enclosed by the fence?
Let L be the length of fence parallel to the river.
A = L*(2400-L)/2 = 1200L - L^2/2
dA/dL = 0 = 1200 - L (using calculus)
L = 1200 m for maximum A
Amax = 1200*600 = 720,000 m^2
You can get the same answer by completing the square.
A = -(1/2)(L^2 -2400L + 1,440,000) + 720,000
= (-1/2)(L-1200)^2 + 720,000
That obviously has a maximum value when L = 1200.
To find the maximum area that can be enclosed by the fence, we need to determine the dimensions of the rectangular area. Let's break it down into steps:
Step 1: Identify the sides to be enclosed by the fencing. In this case, three sides need to be enclosed which means one side will be open to the creek.
Step 2: Let's assume the width of the rectangular area as 'w' and the length as 'l'. Since three sides are enclosed, the perimeter of the rectangular area will be: Perimeter = w + l + w = 2w + l.
Step 3: We are given that the farmer has 2400 meters of fencing material, which means the total length of the fencing material required equals the perimeter: 2400 = 2w + l.
Step 4: We need to express l in terms of w or w in terms of l. Let's rearrange the equation to isolate l: l = 2400 - 2w.
Step 5: Now we can express the area (A) of the enclosed rectangular area in terms of 'w' as: A = w * l = w * (2400 - 2w).
Step 6: To find the maximum area, we need to determine the value of 'w' that optimizes the area. To do this, we can find the vertex of the parabolic function A = w * (2400 - 2w).
Step 7: The formula for the x-coordinate of the vertex of a quadratic function in the form f(x) = ax^2 + bx + c is given by: x = -b / (2a). In this case, 'w' represents the x-coordinate, 'a' is -2 (the coefficient of w^2), and 'b' is 2400.
Step 8: Plugging in the values, we find w = -2400 / (2 * -2) = 600. This gives us the value of 'w' that maximizes the area.
Step 9: Now substitute the value of 'w' back into the equation l = 2400 - 2w to find the corresponding value of 'l': l = 2400 - 2 * 600 = 1200.
Step 10: Finally, calculate the maximum area by substituting the values of 'w' and 'l' into the area formula: A = w * l = 600 * 1200 = 720,000 square meters.
Therefore, the maximum area that can be enclosed by the fence is 720,000 square meters.