A farmer wants to enclose a rectangular area alongside a river using 100 meters of fencing. What dimensions should be chosen to maximize the enclosed area?
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To maximize the enclosed area, we can use the properties of a rectangle, which has two equal sides opposite each other.
Let's assume the width of the rectangle is x. Since there are two equal sides, the length of the rectangle is (100 - 2x). This is because there are two sides with length x, and the remaining fencing length will be used for the other two sides.
The area of the rectangle can be calculated by multiplying its length and width: A = x * (100 - 2x).
To find the dimensions that maximize the enclosed area, we need to find the maximum value of A concerning the variable x. We can do this by analyzing the properties of the quadratic equation representing the area.
The equation can be simplified: A = 100x - 2x^2.
To find the maximum, we take the derivative of A with respect to x and set it equal to zero: dA/dx = 100 - 4x = 0.
Solving for x: 4x = 100 → x = 25.
So, the width of the rectangle is 25 meters.
Substituting this value back into the length equation: length = 100 - 2x = 100 - 2 * 25 = 100 - 50 = 50 meters.
Therefore, the farmer should choose dimensions of 25 meters by 50 meters to maximize the enclosed area.
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