Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

From the early \(1500 \mathrm{~s}\) to nearly 1700, the Turkish town of Iznik was famous for its beautiful colored tiles. In the 1990 s, tile-making was pursued with renewed vigor in the town. In the late \(1990 \mathrm{~s}\), a new mosque was built, and the walls, both inside and outside, are currently being covered with the blue and red tiles for which the town is known. If the mosque cost \(C\) dollars to construct with an additional \(T\) dollars for each tile used, nd the total cost as a function of \(x\), where \(x\) is the number of tiles used.

Step-by-Step Solution

Answer

The total cost as a function of the number of tiles used, \(x\), is given by the equation \(f(x) = C + Tx\).

Step 1: Understand the cost components

In this problem, there are two components of the total cost for the mosque. The first one is the cost of the construction designated as \(C\) dollars and the second is the cost of the tiles used. The cost of the tiles are given by \(T\) dollars for each tile used.

Step 2: Define the cost function

To solve this problem, it is necessary to formulate a cost function factoring in the number of tiles used. This can be done by adding the cost of the construction, \(C\), and then adding the cost of each individual tile, denoted as \(T\), multiplied by the number of tiles used, \(x\).

Step 3: Write the final function

The cost function will be denoted as \(f(x)\), where \(x\) represents the number of tiles used. So the function will be \(f(x) = C + Tx\). Here, \(C\) is a constant that represents the initial construction cost, \(T\) is another constant that represents the cost of each tile, and \(x\) is the variable that represents the number of tiles used. The total cost therefore increases linearly with the number of tiles used.

Key Concepts

Constructing Functions

When faced with real-world scenarios where you need to calculate costs, construct plans, or make predictions, creating a mathematical function is a fundamental approach to quantifying and understanding relationships between variables. In the case of constructing a cost function for building a mosque as mentioned in the exercise, we start by identifying the elements that influence the final cost. These include a fixed cost, in this scenario the construction cost (\(C\) dollars), and a variable cost, here being the cost of tiles (\(T\) dollars per tile).

To construct a function, we look at how each component of cost behaves with the change in another variable, which, for this example, is the number of tiles (\(x\)). Here's the basic principle: for every tile added, the total cost increases by the cost of one tile. Therefore, the cost function we construct reflects a direct relationship between the number of tiles and the total cost, linear in nature, and can be written as \(f(x) = C + Tx\).

While constructing functions, it's important to keep in mind their real-world implications. In teaching, it's useful to encourage students to visualize this process, perhaps by imagining stacking tiles and watching the total cost rise with each additional tile. This tangible representation reinforces the concept that functions are not just abstract mathematical constructs but tools for making sense of our world.

Linear Functions

Linear functions are the simplest type of function we encounter in calculus and algebra. They have the form \(f(x) = mx + b\), where \(m\) is the slope or the rate at which \(y\) changes when \(x\) changes, and \(b\) is the y-intercept, the value of \(y\) when \(x = 0\). These functions graph as straight lines, which is why we call them 'linear'.

In the context of the tile cost problem from the textbook, the cost function \(f(x) = C + Tx\) is a linear function. Here, \(T\) corresponds to the slope \(m\), which indicates how much the cost increases for each additional tile. The fixed construction cost \(C\) corresponds to the y-intercept \(b\), which represents the initial amount paid before any tiles are used.

Understanding linear functions is essential because they are used to model situations where there is a constant rate of change, just like in the cost of adding tiles. Moreover, their simplicity makes them a great starting point for learning more complex functions and mathematical modeling, paving the way for grasping more intricate relationships in mathematics and real-world applications.

Mathematical Modeling

Mathematical modeling involves creating equations or functions to represent real-world systems in a way that we can analyze, make predictions, and solve problems. It turns a scenario with potentially many influencing factors into a simplified, yet accurate, representation that captures the essence of the situation.

In constructing a model, like the cost function for the mosque, the key steps involve identifying variables, understanding how they interact, and encapsulating these relationships in mathematical terms. For the mosque, the cost function models this interaction where the total cost is dependent on the number of tiles and includes both a fixed and a variable component.

The constructed cost function \(f(x) = C + Tx\) is a basic linear model, which, though simple, effectively assists in predicting the total cost for any number of tiles. This foundational model in mathematical modeling serves as a stepping stone for students to progress onto more advanced concepts, such as multivariable calculus or systems of differential equations, which describe a wider range of real-world phenomena. Modeling is not only crucial in the field of mathematics but also in various disciplines like economics, engineering, and the social sciences, making mathematical modeling an indispensable skill in the toolkit of scientists and researchers.