Calculate Brenton's Weekly Pay Based On Hours Worked

Hey guys! Let's dive into how we can figure out Brenton's weekly pay, which is a super interesting problem involving different pay rates for regular hours and overtime. We'll break it down step-by-step so it’s crystal clear. So, Brenton's weekly pay isn't just a flat rate; it changes based on how many hours he clocks in. This makes it a function, a mathematical way of saying the output (his pay) depends on the input (his hours worked). This is a common scenario in many jobs, so understanding how to calculate this is pretty useful.

Understanding the Basics of Brenton's Pay Structure

Okay, so the first key thing to grasp is Brenton's pay structure. He earns $20 for every hour he works up to 40 hours. Think of this as his regular pay. But here’s where it gets interesting: anything over 40 hours is considered overtime, and for those extra hours, he gets a sweet $30 per hour. This is a pretty standard setup in many jobs, designed to compensate employees for putting in extra time and effort. Now, why is this important? Well, it means we can't just multiply his total hours by a single rate. We need to consider two separate scenarios: when he works 40 hours or less, and when he works more than 40 hours. This split is what makes the problem a bit more complex, but don’t worry, we'll tackle it together. Remember, the goal here is to find a formula or a function that accurately calculates his pay P(h) based on the number of hours h he works. This will involve a bit of algebraic thinking, but nothing too scary. We’ll use the information about his regular pay and overtime rates to build this function, making sure it correctly reflects how his earnings change as his work hours increase. This kind of problem is a fantastic way to see how math applies to real-world situations, like figuring out your own paycheck someday!

Defining the Function P(h) for Brenton's Weekly Pay

So, how do we actually write down this function, P(h)? This is where the magic happens, guys! We need to create a mathematical expression that captures both scenarios: regular hours and overtime. Let’s start with the easier part: when Brenton works 40 hours or less. In this case, his pay is simply $20 multiplied by the number of hours he works. We can write this as: P(h) = 20h, when h ≤ 40. This means if he works 20 hours, his pay is 20 * $20 = $400. Simple, right? Now, for the overtime situation, it’s a bit trickier. He still gets paid $20 per hour for the first 40 hours, so that’s a constant $20 * 40 = $800. But for every hour over 40, he gets $30. If he works h hours total, and 40 of those are at the regular rate, then the number of overtime hours is (h - 40). So, his overtime pay is 30 * (h - 40). Adding this to his regular pay, we get the total pay for overtime hours: P(h) = 800 + 30(h - 40), when h > 40. See how we’ve created two different equations, one for each scenario? This is called a piecewise function, and it’s perfect for situations like this where the calculation changes depending on the input. By putting these two equations together, we get a complete picture of how Brenton's weekly pay is determined. This is a powerful tool for understanding and predicting his earnings based on the hours he puts in.

Applying the Function: Examples of Calculating Brenton's Pay

Alright, let's get practical and use our fancy function, P(h), to calculate Brenton's pay in a few different scenarios. This is where things get really interesting because we can see the function in action. First, let’s say Brenton works exactly 40 hours in a week. This falls under our first scenario, where h ≤ 40. Using the equation P(h) = 20h, we get P(40) = 20 * 40 = $800. So, he earns $800 for a 40-hour workweek. Makes sense, right? Now, let’s crank things up a bit. What if Brenton works 45 hours? Now we’re in overtime territory, meaning h > 40. We need to use the second part of our function: P(h) = 800 + 30(h - 40). Plugging in 45 for h, we get P(45) = 800 + 30(45 - 40) = 800 + 30 * 5 = 800 + 150 = $950. He earns a cool $950 for that week! See how the overtime pay bumps up his earnings? Let's try one more example. Imagine Brenton really hustles and works 50 hours in a week. Using the same overtime equation, P(50) = 800 + 30(50 - 40) = 800 + 30 * 10 = 800 + 300 = $1100. Wow, $1100! This clearly shows the impact of overtime hours on his paycheck. By using this function, we can easily determine Brenton's pay for any number of hours he works, which is super handy for both him and his employer.

Graphing the Function P(h) to Visualize Brenton's Earnings

Okay, we've got the equations down, we've done some calculations, but let’s take this a step further and visualize what's going on. Graphing the function P(h) gives us a fantastic way to see how Brenton’s pay changes with the hours he works. Think of it as a picture of his earnings! The graph will have two distinct sections, reflecting our piecewise function. For the first part, when h ≤ 40, the equation is P(h) = 20h. This is a straight line starting at the origin (0,0) and going upwards with a slope of 20. Every hour he works adds $20 to his pay, so the line steadily climbs. At 40 hours, the line reaches the point (40, 800), representing his $800 earnings for a full regular workweek. Now, for the overtime part, where h > 40, the equation changes to P(h) = 800 + 30(h - 40). This is another straight line, but it starts at the point (40, 800) and has a steeper slope of 30. Why steeper? Because he earns $30 for each overtime hour, which is more than his regular rate. This means the line climbs more quickly for hours beyond 40. The graph beautifully illustrates how his pay increases more rapidly once he starts working overtime. By looking at the graph, we can quickly estimate his pay for any given number of hours. It’s a powerful visual tool that makes the relationship between hours worked and earnings crystal clear. Plus, it makes the whole concept of a piecewise function a lot less abstract and much more relatable.

Key Takeaways and the Importance of Piecewise Functions

So, what have we learned, guys? We've dived deep into calculating Brenton's weekly pay, created a piecewise function to represent his earnings, and even visualized it with a graph. The key takeaway here is understanding how piecewise functions work and how they can accurately model real-world situations like this one. Remember, a piecewise function is just a function that has different rules or equations for different intervals of its input. In Brenton's case, the input is the number of hours he works, and the rule changes when he exceeds 40 hours. This kind of function is incredibly versatile. Think about other situations where different rules apply based on certain conditions. For example, tax brackets work similarly: you pay different tax rates on different portions of your income. Or consider shipping costs, which might have a flat rate up to a certain weight and then increase per pound after that. The possibilities are endless! Understanding piecewise functions allows us to model these scenarios mathematically, make predictions, and analyze the outcomes. It’s a valuable tool in mathematics and has tons of practical applications. By breaking down Brenton's pay structure, we’ve not only learned how to calculate his earnings but also gained a deeper appreciation for the power and flexibility of piecewise functions in describing the world around us. This is math in action, folks, and it’s pretty awesome!