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The problem provided in the image involves a pricing structure for a transportation company based on the volume of goods transported. Let me break it down step by step:

### Problem Summary:
1. **Pricing for up to 6 m³**: The company charges $20 for each cubic meter (m³) if the total volume is up to 6 m³.
2. **Discount beyond 6 m³**: For customers who send more than 6 m³, the company offers a 10% discount on the cost of each additional cubic meter beyond 6 m³.

The questions are:
a) Find the function that describes the total revenue of the company based on the total volume of the cargo.
b) Graph the revenue function obtained in part (a).
c) Calculate how much a customer who sent 10 m³ of cargo paid.

### Step a) Finding the Revenue Function

Let:
- $ V $ be the volume of the cargo in cubic meters (m³),
- $ R(V) $ be the total revenue as a function of $ V $.

The revenue function has two cases:
- **Case 1: $ V \leq 6 $** (no discount):
  \[
  R(V) = 20V
  \]
  For volumes up to 6 m³, the cost is $20 per m³.

- **Case 2: $ V > 6 $** (with discount on the extra volume):
  For the first 6 m³, the cost is:
  \[
  20 \times 6 = 120
  \]
  For the volume beyond 6 m³, the cost per m³ is reduced by 10%, which means the cost per m³ becomes:
  \[
  20 \times (1 - 0.10) = 20 \times 0.90 = 18
  \]
  Therefore, for $ V > 6 $, the revenue function is:
  \[
  R(V) = 120 + 18(V - 6)
  \]

Thus, the revenue function is:
\[
R(V) = 
\begin{cases} 
20V & \text{if } V \leq 6 \\
120 + 18(V - 6) & \text{if } V > 6 
\end{cases}
\]

### Step b) Graphing the Revenue Function

The graph of $ R(V) $ is piecewise:
- For $ V \leq 6 $, it's a linear function with slope 20 (straight line starting from the origin).
- For $ V > 6 $, it's another linear function, but with a smaller slope of 18.

This will create a two-segment line where the slope changes at $ V = 6 $.

### Step c) Calculating for 10 m³

For $ V = 10 $, we use the second part of the revenue function (since $ V > 6 $):
\[
R(10) = 120 + 18(10 - 6)
\]
\[
R(10) = 120 + 18 \times 4 = 120 + 72 = 192
\]

So, a customer who sent 10 m³ of cargo would pay **$192**.

---

Would you like more details or have any questions about the steps? Here are some related questions you might find useful:

1. How do you determine the slope in piecewise functions?
2. What is the importance of piecewise functions in real-life applications?
3. How would the revenue function change if the discount were larger, say 20%?
4. Can you explain how to find the intercepts in these types of functions?
5. How would the graph look if the discount applied from the very first cubic meter?

### Tip:
For piecewise functions, always ensure the transition point (in this case, $ V = 6 $) is correctly handled to avoid calculation errors when moving between different function behaviors.

Una empresa de transporte cobra $20 por cada 1 m³ de carga si el volumen total de la carga es de hasta 6 m³. Para incentivar a los clientes que envían cargas grandes, la empresa ofrece un descuento de 10% en el costo por cada 1 m³ adicional cuando el volumen de la carga supera los 6 m³. Encuentra la función de ingresos, grafica, y calcula el pago para 10 m³.

Solve a transportation company's pricing problem with a step-by-step explanation of how to find the revenue function, graph it, and calculate the cost for sending 10 m³ of cargo. This problem involves linear piecewise functions and discount calculations.

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