(a) find a function that gives the area a(x) of the field (in square meters) in terms of x. a(x)= (b) what…

Explanation:

Step1: Assume the length along the river

Let the side - length perpendicular to the river be (x) meters. Suppose the total length of the fence is (L) (not given in the problem, but assume we have a fixed amount of fencing). If we have three sides of length (x) and one side along the river, and let the length of the side along the river be (y). Let's assume we have a total of 1200 meters of fencing. Then (3x + y=1200), so (y = 1200 - 3x). The area of a rectangle (A(x)=x\times y). Substituting (y = 1200 - 3x) into the area formula, we get (A(x)=x(1200 - 3x)=1200x-3x^{2}).

Step2: Find the maximum of the function

The function (A(x)=- 3x^{2}+1200x) is a quadratic function of the form (y = ax^{2}+bx + c), where (a=-3), (b = 1200), and (c = 0). The vertex of a quadratic function (y = ax^{2}+bx + c) has its (x) - coordinate given by (x=-\frac{b}{2a}). Substituting (a=-3) and (b = 1200) into the formula (x=-\frac{b}{2a}), we have (x=-\frac{1200}{2\times(-3)} = 200).

Step3: Calculate the maximum area

Substitute (x = 200) into the area function (A(x)=1200x-3x^{2}). (A(200)=1200\times200-3\times200^{2}=240000-3\times40000=240000 - 120000=120000).

Answer:

(a) (A(x)=1200x - 3x^{2}) (b) 200 (c) 120000