Problem 92 - A Graphical Approach to Precalculus with Limits

Question

If college tuition costs \(\$ 192\) per credit and fees are fixed at \(\$ 275,\) write a formula for a function \(f\) that calculates the tuition and fees for taking \(x\) credits. What is the total cost of taking 11 credits?

Step-by-Step Solution

Answer

The total cost for taking 11 credits is \( \$ 2387 \).

Step 1: Understand the Problem

We need to find a function that represents the total cost of tuition and fees when taking a certain number of credits, where each credit costs \( \\( 192 \) and the fees are fixed at \( \\) 275 \). We also need to calculate what this total cost is when taking 11 credits.

Step 2: Identify the Variables

Let \( x \) represent the number of credits taken. We need to express the total cost \( f(x) \) as a function of \( x \).

Step 3: Formulate the Equation

The cost for \( x \) credits is \( 192x \) (since each credit costs \( \\( 192 \)). Adding the fixed fees of \( \\) 275 \), the formula for the function \( f(x) \) is: \[ f(x) = 192x + 275 \]

Step 4: Substitute and Solve for 11 Credits

To find the total cost of taking 11 credits, substitute \( x = 11 \) into the function: \[ f(11) = 192(11) + 275 \] Calculate the credit cost: \( 192 \times 11 = 2112 \). Add the fixed fees: \( 2112 + 275 \).

Step 5: Compute the Total Cost

Perform the calculation from the previous step: \[ 2112 + 275 = 2387 \] Therefore, the total cost for 11 credits is \( \$ 2387 \).

Key Concepts

Linear Equations in Precalculus

In precalculus, linear equations are fundamental. These equations model relationships where a constant rate of change is involved.
To understand this, think of a straight line, which is the graphical representation of a linear equation. It's called 'linear' because its graph is a line.
A typical linear equation looks like this: \[ y = mx + b \]where:

• \( y \) is the dependent variable, a value depending on \( x \).
• \( m \) is the slope, which tells you how steep the line is. It represents the rate of change, or in other words, how much \( y \) increases for each increase in \( x \).
• \( x \) is the independent variable, the input or the number that you can choose.
• \( b \) is the y-intercept, the point where your line crosses the y-axis.

In the context of tuition cost, we use a similar linear equation format. The cost changes linearly with the number of credit hours.

Cost Analysis

Cost analysis is all about understanding and breaking down the total expenses involved in a scenario. When assessing tuition costs, two main parts can be analyzed:

• Variable Costs: These costs vary depending on the number of credits taken.
  The exercise shows a credit cost of \( \\(192 \) per credit. So, if you take more credits, the cost will be higher.
• Fixed Costs: These are the costs that remain constant, regardless of the number of credits.
  In our example, the fixed fee is \( \\)275\). Whether you take one credit or several, this fee doesn't change.

By understanding both components, you gain a complete picture of what your expenses will look like. It's important for budgeting and making informed financial decisions.

Function Formulation

Creating a function is like crafting a mathematical formula that models a real-world situation.
In our exercise, we want a function that calculates the total cost of tuition based on the number of credits. The process of formulating the function includes the following steps:

• **Identify Variables:** Decide what's changing and what's constant. Here, credits \( x \) vary, while fees are constant.
• **Understand Relationships:** The cost increases with more credits. Each credit has the same cost.
• **Combine Components:** Add together both the variable cost (\(192x\)) and fixed cost (\(\$275\)) to get the total cost.

Together, these steps form \[ f(x) = 192x + 275 \], a formula that represents the tuition cost for any given number of credits \( x \). By plugging different values of \( x \) into the function, you can see how changes in credits affect the overall cost. This is essential for planning how many credits one can afford to take.