To model the number of gallons of fuel oil in the tank after $x$ minutes, we can use a linear function. Here's how we derive it:

Step-by-step derivation:

• Initial amount of fuel oil: 200 gallons
• Rate of filling: 6 gallons per minute

The general formula for a linear function is:

$f(x) = mx + b$

Where:

• $f(x)$ is the number of gallons of fuel in the tank after $x$ minutes,
• $m$ is the rate of change (6 gallons per minute),
• $b$ is the initial amount of fuel in the tank (200 gallons).

Formula for the function:

$f(x) = 6x + 200$

This function models the number of gallons of fuel in the tank after $x$ minutes.

Would you like to go into more details, or do you have any questions? Here are 5 related questions for further practice:

1. How much fuel will be in the tank after 10 minutes?
2. How long will it take for the tank to contain 500 gallons of fuel?
3. If the rate were reduced to 4 gallons per minute, what would the new function be?
4. How many gallons are added in 15 minutes?
5. What is the slope of the function and what does it represent?

Tip: In a linear function, the slope $m$ represents the rate of change, and the y-intercept $b$ represents the initial value at $x = 0$.