A commercial airline has calculated that the approximately fuel mileage for its 600-passenger airplane is 0.2 miles per gallon when the plane travels at an average speed of 500 miles per hour. Flight 818's fuel tank has 42,000 gallons of fuel at the beginning of an international flight. If the plane travels at average speed of 500 miles per hour, which of the following functions f models the number of gallons of fuel remaining in the tank t hours after the flight begins?

Options

A. `f(t)=42,000-(500t)/(0.2)`
B. `f(t)=42,000-(0.2)/(500t)`
C. `f(t)=(42,000-500t)/(0.2)`
D. `f(t)=(42,000-0.2)/(500t)`

Explanation

To solve the problem, we need to determine the function \( f(t) \) that models the number of gallons of fuel remaining in the tank \( t \) hours after the flight begins.

### Step-by-Step Solution:

1. **Identify Initial Fuel**:
The airplane starts with 42,000 gallons of fuel.
\[ \text{Initial Fuel} = 42,000 \text{ gallons} \]

2. **Determine Fuel Mileage**:
The fuel mileage is given as 0.2 miles per gallon. This means for every gallon of fuel, the airplane can travel 0.2 miles.

3. **Calculate Distance Traveled in t Hours**:
The airplane travels at an average speed of 500 miles per hour. Therefore, in \( t \) hours, the distance traveled is:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 500 \times t \text{ miles} \]

4. **Calculate Fuel Used**:
To find out how many gallons of fuel are used for the distance traveled, we can use the fuel mileage. Since the airplane travels 0.2 miles for every gallon, the gallons of fuel used for \( 500t \) miles can be calculated as follows:
\[ \text{Gallons Used} = \frac{\text{Distance}}{\text{Mileage}} = \frac{500t}{0.2} \]
Simplifying this gives:
\[ \text{Gallons Used} = 2500t \]

5. **Calculate Remaining Fuel**:
The remaining fuel in the tank after \( t \) hours can be expressed as:
\[ \text{Remaining Fuel} = \text{Initial Fuel} - \text{Gallons Used} = 42,000 - 2500t \]

6. **Define the Function**:
Thus, the function \( f(t) \) that models the number of gallons of fuel remaining in the tank \( t \) hours after the flight begins is:
\[ f(t) = 42,000 - 2500t \]

### Final Function:
\[ f(t) = 42,000 - 2500t \]