A farmer wants to fence off a rectangular plot of land next to a river. If he has 1200 meters of fencing and the river forms one side of the rectangle, what are the dimensions of the plot that will maximize its area?
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Let's denote the width of the rectangular plot as x. Since the river forms one side of the rectangle, the length of the rectangular plot will be 1200 - x.
The area of the rectangular plot is the product of its length and width. Therefore, the area A(x) is given by:
A(x) = x (1200 - x)
Expanding this expression, we get:
A(x) = 1200x - x^2
To maximize the area, we need to find the value of x that makes A(x) maximum. This can be done by finding the vertex of the quadratic equation A(x).
The vertex of a quadratic equation of the form ax^2 + bx + c is given by (x, y), where x = -b/2a and y = f(x) = ax^2 + bx + c.
In our case, a = -1 (since the coefficient of x^2 is -1), b = 1200, and c = 0. Therefore:
x = -b/2a = -1200/(2*(-1)) = 600
Therefore, the width of the rectangular plot that maximizes its area is 600 meters. The length of the rectangular plot is then 1200 - 600 = 600 meters as well.
Hence, the dimensions of the plot that will maximize its area are 600 meters by 600 meters.
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