Maximum Distance Calculation For Vehicle Fuel Consumption Function Y = -8x + 45

Hey guys! Today, we're diving into a super practical math problem that many of us can relate to – figuring out how far a vehicle can travel before it runs out of gas. We'll be using a simple equation to model fuel consumption and distance, and by the end of this article, you'll be able to calculate the maximum distance your own vehicle (or a hypothetical one) can travel. So, buckle up and let's get started!

Understanding the Fuel Consumption Function

At the heart of our problem is the function y = -8x + 45. Now, what does this mean? Let's break it down. In this equation, 'y' represents the amount of fuel remaining in the vehicle's tank (in some unit, like liters or gallons), and 'x' represents the distance traveled (in kilometers or miles). The equation tells us how the amount of fuel decreases as the distance increases. The key here is the negative slope (-8), which indicates that for every unit of distance traveled, the fuel decreases by 8 units. The '+ 45' is the initial amount of fuel in the tank when the journey begins (when x = 0). This type of equation is called a linear equation, and it's a powerful tool for modeling real-world situations where there's a consistent rate of change.

To truly grasp this, imagine your car's fuel gauge. It starts at a certain level (in this case, 45 units) and gradually decreases as you drive. The function y = -8x + 45 is a mathematical representation of this process. The steeper the slope (the -8 part), the faster the fuel is being consumed. A shallow slope would mean the fuel is being used more slowly. Understanding this relationship between the equation and the real-world scenario is crucial for solving the problem. We're not just crunching numbers; we're translating a mathematical model into a tangible concept. This is what makes math so fascinating – its ability to describe and predict real-life situations. Think about other scenarios where similar equations might be used, like calculating the depreciation of a car's value over time or the cooling of a cup of coffee. Linear functions are all around us!

Now, let's zoom in on the constants and coefficients within our equation. The '-8' isn't just a random number; it's the fuel consumption rate. It tells us exactly how much fuel is burned for each unit of distance traveled. A larger number (in magnitude) would mean higher fuel consumption, while a smaller number would mean better fuel efficiency. The '+45' is our starting point – the initial fuel level. This is important because it sets the limit on how far we can travel. We can't travel further than the fuel in our tank allows! The interplay between the slope and the initial value is what determines the maximum distance we can cover. Consider what would happen if we had a different starting fuel level, say 60 units instead of 45. We'd obviously be able to travel further. Or, imagine a more fuel-efficient vehicle with a gentler slope, like -4. It would cover more distance with the same amount of fuel. These variations highlight the importance of understanding each component of the equation.

Determining the Root (Zero) of the Function

The question asks us to find the maximum distance the vehicle can travel until the tank is empty. In mathematical terms, this means we need to find the point where y (the remaining fuel) equals zero. This point is also known as the root or zero of the function. It's where the graph of the function intersects the x-axis (the distance axis in our case). To find the root, we set y = 0 in our equation and solve for x. So, we have 0 = -8x + 45. Our goal now is to isolate 'x' on one side of the equation.

Solving for 'x' is a fundamental algebraic skill. We start by adding 8x to both sides of the equation to get 8x = 45. This moves the term with 'x' to the left side, making it positive and easier to work with. Then, we divide both sides by 8 to isolate 'x'. This gives us x = 45/8. This fraction represents the distance the vehicle can travel before running out of fuel. But what does 45/8 actually mean in practical terms? To understand this better, we can convert it into a decimal. Dividing 45 by 8 gives us 5.625. So, the vehicle can travel 5.625 units of distance. If our distance is measured in kilometers, this means the vehicle can travel 5.625 kilometers before the fuel tank is empty. If it's miles, then it's 5.625 miles. The key is to remember the units of measurement to give the answer context.

Let's think about the significance of this root in the context of our fuel consumption scenario. It represents the maximum distance the vehicle can travel on a full tank of fuel, according to our mathematical model. Anything beyond 5.625 units, and the vehicle will run out of gas. This is a crucial piece of information for planning a journey. You wouldn't want to embark on a trip longer than 5.625 units without refueling! This practical application is what makes understanding these mathematical concepts so valuable. We can use them to make informed decisions in our daily lives. Imagine using this same concept to plan a road trip, ensuring you have enough fuel to reach your destination or a refueling point. Or, think about how companies use similar models to optimize fuel consumption in their vehicle fleets.

Calculating the Maximum Distance

Now that we've found the root, x = 5.625, we know the maximum distance the vehicle can travel. This value represents the point where the fuel tank is completely empty. To reiterate, this distance is 5.625 units, where the unit depends on the context of the problem (kilometers, miles, etc.). It's crucial to include the units in your final answer to provide a complete and meaningful solution. Simply saying