The problem describes a scenario where three children, Soufyane, Kadafi, and Maxwell, are tasked with filling water containers. Soufyane uses a cylinder, Kadafi uses a truncated pyramid, and Maxwell uses a truncated cone. We need to calculate how much money each child will receive if they fill their container completely, given that each 10 liters of water earns them 75 FCFA. We are given the dimensions of each container: - Soufyane: Cylinder with height $h_1 = 2m$ and radius $r_1 = 1m$. - Kadafi: Truncated pyramid formed from a regular pyramid, cut at one-third of the original height from the base. The original pyramid has height $h_2 = 4m$ and a square base with side length $2m$. - Maxwell: Truncated cone formed from a cone, cut at one-third of the original height from the base. The original cone has height $h_3 = 4m$ and base radius $r_3 = 1.5m$. We are given that $\pi = 3.14$.

1. Problem Description

The problem describes a scenario where three children, Soufyane, Kadafi, and Maxwell, are tasked with filling water containers. Soufyane uses a cylinder, Kadafi uses a truncated pyramid, and Maxwell uses a truncated cone. We need to calculate how much money each child will receive if they fill their container completely, given that each 10 liters of water earns them 75 FCFA.

We are given the dimensions of each container:

- Soufyane: Cylinder with height $h_1 = 2m$ and radius $r_1 = 1m$.

- Kadafi: Truncated pyramid formed from a regular pyramid, cut at one-third of the original height from the base. The original pyramid has height $h_2 = 4m$ and a square base with side length $2m$.

- Maxwell: Truncated cone formed from a cone, cut at one-third of the original height from the base. The original cone has height $h_3 = 4m$ and base radius $r_3 = 1.5m$.

We are given that $\pi = 3.14$.

2. Solution Steps

First, we calculate the volume of each container. Remember that 1 cubic meter is equal to 1000 liters.

*Soufyane (Cylinder)*

The volume of a cylinder is given by $V = \pi r^2 h$.

$V_1 = \pi r_1^2 h_1 = 3.14 \times (1m)^2 \times 2m = 6.28 m^3$.

In liters, this is $6.28 \times 1000 = 6280$ liters.

The number of 10-liter buckets is $6280 / 10 = 628$.

Soufyane receives $628 \times 75 FCFA = 47100 FCFA$.

*Kadafi (Truncated Pyramid)*

The original pyramid has height $h_2 = 4m$ and base side length $a = 2m$.

The volume of the original pyramid is $V = \frac{1}{3} \times base \times height = \frac{1}{3} \times a^2 \times h_2 = \frac{1}{3} \times (2m)^2 \times 4m = \frac{16}{3} m^3$.

The smaller pyramid that is cut off has height $h' = \frac{2}{3}h_2 = \frac{8}{3}m$. By similar triangles, the side length of the smaller pyramid's base $a'$ satisfies

$\frac{a'}{a} = \frac{h'}{h_2} = \frac{\frac{8}{3}}{4} = \frac{2}{3}$. Thus $a' = \frac{2}{3} a = \frac{2}{3} \times 2m = \frac{4}{3} m$.

The volume of the smaller pyramid is $V' = \frac{1}{3} \times (a')^2 \times h' = \frac{1}{3} \times (\frac{4}{3}m)^2 \times \frac{8}{3}m = \frac{1}{3} \times \frac{16}{9} \times \frac{8}{3} m^3 = \frac{128}{81} m^3$.

The volume of the truncated pyramid is $V_2 = V - V' = \frac{16}{3} - \frac{128}{81} = \frac{16 \times 27 - 128}{81} = \frac{432 - 128}{81} = \frac{304}{81} m^3 \approx 3.753 m^3$.

In liters, this is approximately $3.753 \times 1000 = 3753$ liters.

The number of 10-liter buckets is approximately $3753 / 10 = 375.3$. Since we can only have whole buckets, we consider 375 buckets.

Kadafi receives $375 \times 75 FCFA = 28125 FCFA$.

*Maxwell (Truncated Cone)*

The original cone has height $h_3 = 4m$ and base radius $r_3 = 1.5m$.

The volume of the original cone is $V = \frac{1}{3} \pi r_3^2 h_3 = \frac{1}{3} \times 3.14 \times (1.5m)^2 \times 4m = \frac{1}{3} \times 3.14 \times 2.25 \times 4 m^3 = 9.42 m^3$.

The smaller cone that is cut off has height $h' = \frac{2}{3}h_3 = \frac{8}{3} m$. By similar triangles, the radius of the smaller cone's base $r'$ satisfies

$\frac{r'}{r_3} = \frac{h'}{h_3} = \frac{\frac{8}{3}}{4} = \frac{2}{3}$. Thus $r' = \frac{2}{3} r_3 = \frac{2}{3} \times 1.5m = 1m$.

The volume of the smaller cone is $V' = \frac{1}{3} \pi (r')^2 h' = \frac{1}{3} \times 3.14 \times (1m)^2 \times \frac{8}{3} m = \frac{1}{3} \times 3.14 \times \frac{8}{3} m^3 = \frac{25.12}{9} m^3 \approx 2.791 m^3$.

The volume of the truncated cone is $V_3 = V - V' = 9.42 - 2.791 = 6.629 m^3$.

In liters, this is approximately $6.629 \times 1000 = 6629$ liters.

The number of 10-liter buckets is approximately $6629 / 10 = 662.9$. Since we can only have whole buckets, we consider 662 buckets.

Maxwell receives $662 \times 75 FCFA = 49650 FCFA$.

3. Final Answer

1. Soufyane should receive 47100 FCFA.

2. Kadafi should receive 28125 FCFA.

3. Maxwell should receive 49650 FCFA.