Answer
$x_{maximum} = 50$; $f(50)=5,000$
Work Step by Step
*Reference back to page 340 for a step-by-step solution to finding maximum and minimum values of quadratic functions.*
For this exercise, the first step is to identify what value the question seeks (maximum or minimum) and to write a function that expresses the variable in question in a quadratic function:
1) Since we seek to maximize the AREA of the plot, we use the formula for area of rectangles (length x width) to construct the following formula:
$$Area Of Plot = f(x) = (200 - 2x)(x) = 200x - 2x^{2}$$
Which we can re-write into a quadratic form $f(x) = ax^{2} + bx + c$:
$$f(x) = -2x^{2} + 200x + 0$$
2) Since we know that, when $a<0$, the maximum of the function can be determined when $x=\frac{-b}{2a}$, we can calculate it as follows:
$$x = \frac{-(200)}{(2\times (-2))} = \frac{-200}{-4} = 50$$
3) Knowing the value of $x$ where the function is maximized, we can now calculate the value of the area of the plot:
$$f(x_{maximum}) = -2x^{2} + 200x$$
$$f(50) = -2(50^{2}) + 200(50)$$
$$f(50) = -2(2,500) + 10,000 = -5,000 + 10,000 = 5,000$$
