Question

(b) a(x)=1300x - 2x²
(c) first write the expression for the derivative used to find the x value that maximizes area.
\frac{da}{dx}=1300 - 4x
the x - value leading to the maximum area is 325
(d) the maximum area of the rectangular plot is 2

Explanation:

Step1: Differentiate the area function

Given \(A(x) = 1300x - 2x^{2}\). Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\), we have \(\frac{dA}{dx}=\frac{d}{dx}(1300x)-\frac{d}{dx}(2x^{2})\).

$$
LATEXBLOCK0
$$

Step2: Find the critical point

Set \(\frac{dA}{dx}=0\) (since at maximum or minimum of a function \(y = f(x)\), \(f^{\prime}(x)=0\)).

$$
LATEXBLOCK1
$$

Step3: Find the maximum area

Substitute \(x = 325\) into \(A(x)=1300x-2x^{2}\).

$$
LATEXBLOCK2
$$

Answer:

The derivative in part (c) is \(\frac{dA}{dx}=1300 - 4x\). The \(x\) - value in part (a) is \(x = 325\). The maximum area in part (d) is \(211250\space m^{2}\)