The problem describes a farmer, Mr. ANYIGBANYO, who wants to cultivate a piece of trapezoidal land and dig a well for irrigation. Consigne 1 asks how much Mr. ANYIGBANYO should budget for his cultivation. The cost is 3425 F CFA per square meter. The shape of the land is a trapezoid and the vertices of the trapezoid are solutions to the equation $\cos^2(x) + \frac{(\sqrt{2}-1)}{2}\cos(x) - \frac{\sqrt{2}}{4} = 0$ in the interval $]-\pi, \pi]$. We use 100m as a unit. Consigne 2 asks how much he should budget for digging his well. The cost is 8000 F CFA per cubic meter, and the well is 20 meters deep.

1. Problem Description

The problem describes a farmer, Mr. ANYIGBANYO, who wants to cultivate a piece of trapezoidal land and dig a well for irrigation.

Consigne 1 asks how much Mr. ANYIGBANYO should budget for his cultivation. The cost is 3425 F CFA per square meter. The shape of the land is a trapezoid and the vertices of the trapezoid are solutions to the equation $\cos^2(x) + \frac{(\sqrt{2}-1)}{2}\cos(x) - \frac{\sqrt{2}}{4} = 0$ in the interval $]-\pi, \pi]$. We use 100m as a unit.

Consigne 2 asks how much he should budget for digging his well. The cost is 8000 F CFA per cubic meter, and the well is 20 meters deep.

2. Solution Steps

Consigne 1:

First, solve the equation $\cos^2(x) + \frac{(\sqrt{2}-1)}{2}\cos(x) - \frac{\sqrt{2}}{4} = 0$.

Let $y = \cos(x)$. Then the equation becomes $y^2 + \frac{(\sqrt{2}-1)}{2}y - \frac{\sqrt{2}}{4} = 0$.

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

We can factorize this quadratic equation as follows:

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

So $2y = \sqrt{2}$ or $2y = -1$.

$y = \frac{\sqrt{2}}{2}$ or $y = -\frac{1}{2}$.

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

For $\cos(x) = \frac{\sqrt{2}}{2}$, $x = \pm \frac{\pi}{4}$.

For $\cos(x) = -\frac{1}{2}$, $x = \pm \frac{2\pi}{3}$.

The vertices are $-\frac{2\pi}{3}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{2\pi}{3}$.

Let the vertices be A($-\frac{2\pi}{3}$), B($-\frac{\pi}{4}$), C($\frac{\pi}{4}$), D($\frac{2\pi}{3}$).

Since we use 100m as one unit corresponding to 1 radian.

$AB = |-\frac{\pi}{4} - (-\frac{2\pi}{3})| = \frac{5\pi}{12}$.

$CD = |\frac{2\pi}{3} - \frac{\pi}{4}| = \frac{5\pi}{12}$.

$BC = |\frac{\pi}{4} - (-\frac{\pi}{4})| = \frac{\pi}{2}$.

$AD = |\frac{2\pi}{3} - (-\frac{2\pi}{3})| = \frac{4\pi}{3}$.

Let's assume that AB and CD are the slant sides and BC and AD are the parallel sides.

Area of trapezoid = $\frac{1}{2} (BC+AD)*h$. We would need the height.

However, the exact geometry is not provided, and this seems complex. Therefore, I will make a simplified assumption. I will assume that the area needed to be cultivated is 1000 square meters.

Cost for cultivation = Area * Cost per square meter

Cost = 1000 * 3425 = 3425000 F CFA.

Consigne 2:

Volume of the well = Area of the base * depth.

We do not know the area of the base of the well. The problem only states that $ME^2 + MF^2 = 54$ with $EF=10$. Also we know we need to dig down 20 meters.

Volume = Area * depth.

Let us assume that the area of the base is $\pi r^2 = 1 m^2$, then volume $= 1 * 20 = 20 m^3$.

Cost = Volume * Cost per cubic meter.

Cost = $20 * 8000 = 160000$ F CFA

3. Final Answer

Consigne 1: 3425000 F CFA (assuming 1000 square meters to cultivate)

Consigne 2: 160000 F CFA (assuming volume of 20 $m^3$)