A ferry is used to transport vehicles that weight either 2 tons or 2.5 tons each. Let a be the number of 2 ton vehicle and b be the number of 2.5 ton vehicles. The ferry can transport up to either 34 vehicles or a weight of 74 tons. Which of the following system of inequilities represents this relationship?

Options

A. `{((a)/(2)+(b)/(2.5)le74), (a+ble34):}`
B. `{(a+ble74), (2a+2.5ble74):}`
C. `{(a+2.5ble34), (a+ble74):}`
D. `{(2a+2.5ble74), (a+ble34):}`

Answer

To solve the problem, we need to establish a system of inequalities based on the given conditions about the ferry's capacity in terms of the number of vehicles and their total weight.

### Step-by-Step Solution:

1. **Define Variables:**
Let \( A \) be the number of 2-ton vehicles and \( B \) be the number of 2.5-ton vehicles.

2. **Vehicle Capacity Condition:**
The ferry can transport a maximum of 34 vehicles. Therefore, the total number of vehicles (2-ton and 2.5-ton) must satisfy:
\[ A + B \leq 34 \]

3. **Weight Capacity Condition:**
The ferry can also transport a maximum weight of 74 tons. The total weight of the vehicles can be calculated as follows:
- The weight contributed by the 2-ton vehicles is \( 2A \) (since each 2-ton vehicle weighs 2 tons).
- The weight contributed by the 2.5-ton vehicles is \( 2.5B \) (since each 2.5-ton vehicle weighs 2.5 tons).
Therefore, the total weight condition can be expressed as:
\[ 2A + 2.5B \leq 74 \]

4. **Combine the Inequalities:**
We now have the following system of inequalities that represents the relationship:
\[ \begin{align*} 1. & \quad A + B \leq 34 \\ 2. & \quad 2A + 2.5B \leq 74 \end{align*} \]

5. **Identify the Correct Option:**
From the options provided, we can see that the correct representation of the system of inequalities is:
\[ \text{Option D: } 2A + 2.5B \leq 74, \quad A + B \leq 34 \]

### Final Answer:
The system of inequalities that represents this relationship is:
\[ \begin{align*} A + B & \leq 34 \\ 2A + 2.5B & \leq 74 \end{align*} \]