The problem describes a school planning to renovate its facilities, including a football stadium, handball stadium, athletics track, and amphitheater. Several sub-problems are presented. We need to find the total budget and the number of square meters of carpet to use.

1. Problem Description

The problem describes a school planning to renovate its facilities, including a football stadium, handball stadium, athletics track, and amphitheater. Several sub-problems are presented. We need to find the total budget and the number of square meters of carpet to use.

2. Solution Steps

First, we will break down the total budget calculation into separate parts, and calculate each part's cost.

Football Stadium:

The stadium's area is not directly provided. It only mentions the solutions of trigonometric equations being related to it. The cost to prevent the lawn from being submerged using sand and cement must be calculated.

The price of sand is 600 FCFA per 15 liters, which covers 0.5 $m^2$.

The price of cement is 3900 FCFA per bag, which covers 3 $m^2$.

Volleyball Stadium:

The volleyball stadium is represented by three points D, E, and F. The area of this triangle DEF is k $m^2$, and k is determined by the equation $k + \sqrt{k^2 + k - 6} = 2$.

Athletics Track:

The athletics track is a triangle ABC with side length 10 m. We need to protect the track by planting advertising panels. Two feet of the panels cover 0.15 m, and each foot costs 750 FCFA. We need more information to calculate the length of the track.

Amphitheater:

The office in the amphitheater is covered with carpet, but the carpet is offered free of charge, so it doesn't affect the total budget.

Solving for k:

$k + \sqrt{k^2 + k - 6} = 2$

$\sqrt{k^2 + k - 6} = 2 - k$

Squaring both sides:

$k^2 + k - 6 = 4 - 4k + k^2$

$5k = 10$

$k = 2$

So, the area of the volleyball stadium is 2 $m^2$. Cost is 30200 FCFA/m^2 so total cost is 2*30200 = 60400 FCFA.

Total Budget:

We don't have enough information from the provided text to find the exact costs for the football stadium and the athletics track. We can however calculate the volleyball court.

Cost of volleyball court= 60400 FCFA.

We know that the carpet to cover the Amphitheater is free of charge.

Consigne 2:

The number of square meters of carpet used is to cover the office in the amphitheater, which corresponds to the set M described by $x = 10cos^2(6\theta) - 6$ and $y = 10sin(6\theta)cos(6\theta)$. Since we can not find this specific number from the provided context, we can say it is at most $6R$.

3. Final Answer

Consigne 1: We cannot determine the total budget from the text given, but we can state that it includes: The sand and cement to prevent lawn submersion in the football stadium, the cost of the volleyball court being 60400 FCFA, and the advertising panels along the athletics track.

Consigne 2: The number of square meters of carpet to use is unknown, but at most it is $6R$.