The problem describes a farmer, Mr. Anyigbanyo, who wants to cultivate a trapezoidal field. First, we need to determine the amount he needs to spend on seeds for his crop. The vertices of the trapezoid are the solutions in $]-\pi, \pi]$ of the equation $cos^2(x) + \frac{\sqrt{2}-1}{2}cos(x) - \frac{\sqrt{2}}{4} = 0$. We know that 1 "unit" equals 100 meters and that one square meter of seed costs 3425 F CFA. Second, we need to find out how much it will cost to dig a well. Initially, the technician's price is 8000 F CFA per cubic meter.

1. Problem Description

The problem describes a farmer, Mr. Anyigbanyo, who wants to cultivate a trapezoidal field. First, we need to determine the amount he needs to spend on seeds for his crop. The vertices of the trapezoid are the solutions in $]-\pi, \pi]$ of the equation $cos^2(x) + \frac{\sqrt{2}-1}{2}cos(x) - \frac{\sqrt{2}}{4} = 0$. We know that 1 "unit" equals 100 meters and that one square meter of seed costs 3425 F CFA. Second, we need to find out how much it will cost to dig a well. Initially, the technician's price is 8000 F CFA per cubic meter.

2. Solution Steps

First, we solve the equation for $cos(x)$.

$cos^2(x) + \frac{\sqrt{2}-1}{2}cos(x) - \frac{\sqrt{2}}{4} = 0$

Let $y = cos(x)$. The equation becomes:

$y^2 + \frac{\sqrt{2}-1}{2}y - \frac{\sqrt{2}}{4} = 0$

Multiply by 4 to simplify:

$4y^2 + 2(\sqrt{2}-1)y - \sqrt{2} = 0$

We can use the quadratic formula to solve for $y$:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

$y = \frac{-2(\sqrt{2}-1) \pm \sqrt{(2(\sqrt{2}-1))^2 - 4(4)(-\sqrt{2})}}{2(4)}$

$y = \frac{-2\sqrt{2}+2 \pm \sqrt{4(2 - 2\sqrt{2} + 1) + 16\sqrt{2}}}{8}$

$y = \frac{-2\sqrt{2}+2 \pm \sqrt{12 - 8\sqrt{2} + 16\sqrt{2}}}{8}$

$y = \frac{-2\sqrt{2}+2 \pm \sqrt{12 + 8\sqrt{2}}}{8}$

$y = \frac{-2\sqrt{2}+2 \pm \sqrt{(2\sqrt{2} + 2)^2}}{8}$

$y = \frac{-2\sqrt{2}+2 \pm (2\sqrt{2}+2)}{8}$

So we have two possible solutions for $y$:

$y_1 = \frac{-2\sqrt{2}+2 + 2\sqrt{2}+2}{8} = \frac{4}{8} = \frac{1}{2}$

$y_2 = \frac{-2\sqrt{2}+2 - 2\sqrt{2}-2}{8} = \frac{-4\sqrt{2}}{8} = -\frac{\sqrt{2}}{2}$

Since $y = cos(x)$, we have $cos(x) = \frac{1}{2}$ and $cos(x) = -\frac{\sqrt{2}}{2}$.

For $cos(x) = \frac{1}{2}$, the solutions in $]-\pi, \pi]$ are $x = \frac{\pi}{3}$ and $x = -\frac{\pi}{3}$.

For $cos(x) = -\frac{\sqrt{2}}{2}$, the solutions in $]-\pi, \pi]$ are $x = \frac{3\pi}{4}$ and $x = -\frac{3\pi}{4}$.

So the vertices of the trapezoid are $A(-\frac{3\pi}{4}), B(-\frac{\pi}{3}), C(\frac{\pi}{3}), D(\frac{3\pi}{4})$.

Since we are taking $100$ meters to be a unit, the $x$-coordinates have to be multiplied by $100$.

Let us consider the trapezoid as made of a rectangle and 2 triangles. The parallel sides are $BC = \frac{2\pi}{3}$. The other parallel side is $AD = \frac{6\pi}{4} = \frac{3\pi}{2}$.

Then the length of the parallel sides are $100 \times \frac{2\pi}{3}$ and $100 \times \frac{3\pi}{2}$. The height of the trapezoid is the distance from the x axis = 0, i.e., $h = 0$. Area of trapezoid $= (1/2) \times (\frac{2\pi}{3} + \frac{3\pi}{2}) \times h$. Since all $y = 0$, the vertices lie on a line. Hence the area is 0, and the amount needed is 0 F CFA. However, based on the fact that the question indicates the area is not zero, and since all y values = 0, there must be something wrong. The correct assumption should be we simply plot the $x$ values on a number line and they represent the $x$ values of the point, therefore, they have $y = 0$.

The area can be considered as the segment from $-\frac{3\pi}{4}$ to $\frac{3\pi}{4}$, hence length $= 2*\frac{3\pi}{4}$ = $\frac{3\pi}{2}$, The area is from $-\frac{\pi}{3}$ to $\frac{\pi}{3}$ length = $\frac{2\pi}{3}$.

Assume the area calculation of the land as: $(\frac{3\pi}{2} - \frac{2\pi}{3}) *100 * 100$. In this case, area = $(\frac{9\pi - 4\pi}{6}) *100 * 100 = \frac{5\pi}{6} *10000$.

Then, the amount to be paid = $\frac{5\pi}{6} *10000 * 3425 \approx 89763693.48$. This leads to the idea that perhaps the question wants the area of the trapezoid formed from $A, B, C, D$ with $x$ values being the $x$ coordinate and $y = 0$. In this case, the area is indeed zero.

Now, consider the second question: The volume of the well is the area of the base (assumed as a circle) multiplied by the depth. The volume of the excavated earth is given as $V = \pi r^2 h$. But they are not asking for radius, but rather give the cost of excavation per $m^3$ as $8000$ F CFA. Given the well goes down 20 m in depth, volume = V. The cost = $8000 * V$. If $V = 1$, cost = 8000 F CFA. If $V = 2$, cost = 16000 F CFA. Let us assume $V = 20$. Cost = 160000 F CFA. I am unable to determine the volume needed for cultivation in any case.

3. Final Answer

For the first question: 0 F CFA

For the second question: It costs 8000 F CFA per cubic meter to dig the well. Since the depth is 20 meters, if the volume excavated is $V$ cubic meters, then the total cost is $8000V$ F CFA. It is impossible to determine the value of V.