A certain jet plane has an average speed of 500 km per hour. It can carry sufficient fuel for a 5 hour flight.
Is this relation a function?

Answer

To determine whether the relation described in the question is a function, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( d \) be the distance traveled by the jet plane.
- \( t \) be the time of the flight in hours.

### Step 2: Use the Speed Formula
We know that speed is defined as:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
Given that the average speed of the jet plane is 500 km/h, we can express this relationship mathematically:
\[ 500 = \frac{d}{t} \]

### Step 3: Rearrange the Equation
From the equation above, we can solve for distance \( d \):
\[ d = 500t \]

### Step 4: Define the Relation
Now we can define the relation \( R \) as follows:
\[ R = \{(d, t) | d = 500t, \, t \in \mathbb{R}, \, 0 < t < 5\} \]
This means that the relation consists of pairs of distance and time, where time \( t \) is a real number between 0 and 5 hours.

### Step 5: Check for Functionality
To check if this relation is a function, we need to see if for every input \( t \) (time), there is a unique output \( d \) (distance).

- For any value of \( t \) in the interval \( (0, 5) \), we can calculate \( d \) using the formula \( d = 500t \).
- Since the equation \( d = 500t \) gives a unique distance for each time \( t \), there cannot be two different distances for the same time.

### Conclusion
Since every input \( t \) corresponds to exactly one output \( d \), we conclude that the relation is indeed a function.

**Final Answer: Yes, this relation is a function.**

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