Skateboard Production Cost: A Mathematical Analysis

Alright guys, let's dive into a super interesting problem involving skateboard production! We're going to explore how math can help us understand the relationship between production hours, the number of skateboards made, and the total manufacturing cost. This is a practical application of functions, and by the end of this article, you'll have a clear understanding of how to tackle such problems. So, grab your thinking caps, and let's get started!

Understanding the Production Function

First, let's break down the production function $f(h) = 325h$. This function tells us how many skateboards a company can produce based on the number of hours, $h$, they dedicate to production. The number 325 is super important here; it represents the rate of production. In simpler terms, for every hour of work, the company can produce 325 skateboards.

To really grasp this, imagine you're managing this skateboard company. If you allocate 1 hour ($h = 1$), you'll get $f(1) = 325 * 1 = 325$ skateboards. Now, what if you decide to work for 5 hours ($h = 5$)? Then, $f(5) = 325 * 5 = 1625$ skateboards. See how straightforward it is? The more hours you put in, the more skateboards you produce, and this relationship is directly proportional, thanks to that constant rate of 325. Understanding this linear relationship is key to planning and forecasting production. If you have a target number of skateboards to produce, you can easily calculate the number of hours needed. For instance, if you need to produce 3250 skateboards, you would solve the equation $3250 = 325h$ for $h$, which gives you $h = 10$ hours. This simple equation empowers you to make informed decisions about resource allocation and production schedules. Moreover, you can use this function to evaluate the impact of increasing or decreasing production hours. For example, if you increase the working hours by 2, you can quickly determine the increase in skateboard production by calculating $325 * 2 = 650$ skateboards. This allows for agile adjustments to meet changing market demands or to optimize production efficiency. The production function $f(h) = 325h$ is not just a mathematical equation; it's a powerful tool that provides insights into the company's operational capabilities and enables data-driven decision-making.

Analyzing the Manufacturing Cost Function

Next, let's look at the manufacturing cost function $g(b) = 0.008b^2 + 8b + 100$. This function tells us the total cost to manufacture $b$ skateboards. It's a bit more complex than the production function because it includes a quadratic term ($0.008b^2$), a linear term ($8b$), and a constant term ($100$). Let’s break each of these down to see what they mean.

The $0.008b^2$ term represents the variable costs that increase at an increasing rate as the number of skateboards produced ($b$) increases. This could be due to things like needing to buy more expensive materials as you scale up production, or increased wear and tear on your equipment. It means that the cost per skateboard increases slightly as you produce more due to inefficiencies or rising material costs. The $8b$ term represents costs that increase linearly with the number of skateboards produced. This could be the cost of labor, where each skateboard requires a certain amount of worker time, and the total labor cost increases in direct proportion to the number of skateboards. For each additional skateboard, you're paying an extra $8 in labor and materials. The constant term, $100, represents fixed costs. These are costs that you have to pay regardless of how many skateboards you produce. This could be rent for your factory, insurance, or the cost of essential equipment. Even if you produce zero skateboards, you still have to pay these $100 in fixed costs. Understanding this cost function is vital for cost management and profitability analysis. For example, if you manufacture 100 skateboards, the total cost would be $g(100) = 0.008(100)^2 + 8(100) + 100 = 0.008(10000) + 800 + 100 = 80 + 800 + 100 = 980$. Therefore, the total cost to produce 100 skateboards is $980. By analyzing this function, you can determine the most cost-effective production level. Additionally, you can use the cost function to evaluate the impact of changes in production volume on total costs. For instance, if you increase production from 100 to 200 skateboards, you can calculate the additional cost and assess whether the increase in revenue justifies the additional expense.

Combining the Functions: Cost as a Function of Hours

Now, the fun part! We want to find an expression that represents the total manufacturing cost as a function of the number of hours, $h$. In other words, we want to find a function that tells us how much it will cost to manufacture skateboards based on the number of hours spent producing them. To do this, we need to combine our two functions, $f(h)$ and $g(b)$. The key here is to recognize that the number of skateboards produced, $b$, is related to the number of hours, $h$, through the function $f(h)$. So, we can substitute $f(h)$ into $g(b)$ to get the total cost as a function of $h$. Mathematically, this means we want to find $g(f(h))$.

We know that $f(h) = 325h$ and $g(b) = 0.008b^2 + 8b + 100$. So, we replace $b$ in $g(b)$ with $325h$: $g(f(h)) = 0.008(325h)^2 + 8(325h) + 100$. Now, let’s simplify this expression. First, we calculate $(325h)^2 = 325^2 * h^2 = 105625h^2$. Then, we multiply this by $0.008$ to get $0.008 * 105625h^2 = 845h^2$. Next, we calculate $8 * 325h = 2600h$. So, our expression becomes: $g(f(h)) = 845h^2 + 2600h + 100$. This new function, $g(f(h))$, tells us the total manufacturing cost as a function of the number of hours, $h$. For example, if the company spends 2 hours on production, the total cost would be: $g(f(2)) = 845(2)^2 + 2600(2) + 100 = 845 * 4 + 5200 + 100 = 3380 + 5200 + 100 = 8680$. Therefore, it would cost $8680 to manufacture skateboards if the company spends 2 hours on production. This composite function allows you to directly relate production hours to total manufacturing costs, making it a powerful tool for budgeting and financial planning.

Practical Applications and Insights

So, what can we do with this function $g(f(h)) = 845h^2 + 2600h + 100$? Well, a lot! This function allows us to analyze how changes in production hours affect the total manufacturing cost. For example, we can use it to determine the optimal number of hours to work to minimize costs or to predict the cost of meeting a certain production target. One practical application is cost optimization. Managers can use this function to find the most cost-effective number of hours to operate. By analyzing the function, they can identify at what point the costs start to increase disproportionately to the number of skateboards produced. This helps in setting production targets that maximize efficiency and minimize waste. Another important application is in budgeting and financial planning. With this function, the company can accurately predict manufacturing costs for different production scenarios. This is crucial for creating realistic budgets and securing financing. For instance, if the company needs to fulfill a large order, they can use the function to estimate the total cost and ensure they have enough resources. Furthermore, the function can be used for break-even analysis. By setting the cost function equal to the revenue function, the company can determine the number of hours needed to break even. This provides valuable insights into the profitability of the production process and helps in making strategic decisions about pricing and marketing. Additionally, this function helps in scenario planning. The company can use it to model different production scenarios and assess the potential impact on costs. For example, they can analyze the effects of increasing or decreasing production hours, changing labor costs, or implementing new technologies. This enables them to make informed decisions and adapt quickly to changing market conditions. In essence, $g(f(h)) = 845h^2 + 2600h + 100$ is more than just a mathematical expression; it's a valuable tool that provides a comprehensive understanding of the relationship between production hours and manufacturing costs, enabling better decision-making and strategic planning.

Conclusion

Wrapping it up, we've seen how to combine two functions to create a new function that gives us valuable information about skateboard production costs. By understanding the production function $f(h)$ and the cost function $g(b)$, we were able to create a composite function $g(f(h))$ that tells us the total manufacturing cost as a function of the number of hours. This kind of analysis is super useful in real-world business scenarios for planning, budgeting, and optimizing production. Keep practicing with these concepts, and you'll become a math whiz in no time! Understanding the interplay between different functions is a fundamental skill that extends beyond the realm of mathematics, empowering you to tackle complex problems and make informed decisions in various fields. Whether you're managing a skateboard company or pursuing other endeavors, the ability to model and analyze relationships using functions is invaluable.