Problem 93 - A Graphical Approach to College Algebra

Question

Tuition and Fees 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: Understanding the Problem

We need to find a formula for calculating the cost of college tuition and fees. The tuition costs \(192 per credit, and there is a fixed fee of \)275. We want to express the total cost as a function of the number of credits,\(x\).

Step 2: Constructing the Function Formula

The cost per credit is \(192, so for \(x\) credits, the tuition will be \(192x\). In addition, there is a fixed fee of \)275. Thus, the function formula expressing the total cost \(f(x)\) is: \[f(x) = 192x + 275\].

Step 3: Calculating the Total Cost for 11 Credits

Now that we have the function \(f(x) = 192x + 275\), substitute \(x = 11\) to find the total cost. This gives: \[f(11) = 192 \times 11 + 275\].

Step 4: Performing the Calculation

Calculate \(192 \times 11 = 2112\). Then, add the fixed fee: \(2112 + 275 = 2387\).

Key Concepts

Function Formula

When tasked with creating a function formula for tuition and fees, it's all about simplifying how costs scale with coursework. Here, the college charges an amount per credit plus a fixed fee. To capture this in a formula:

• The cost per credit is \(192\).
• The fixed fee is a one-time charge of \(275\).

The goal is to create a relation where the total cost depends on the number of credits, represented by \(x\). This gives us the formula: \[f(x) = 192x + 275\]It embodies both variable and fixed components. The term \(192x\) reflects costs that grow with more credits, while \(275\) remains constant regardless of credits taken.

Credits Calculation

Calculating credits is essential for budgeting your academic term since each credit incurs a cost. For this problem, we utilized a formula already derived: \[f(x) = 192x + 275\]To find the total tuition and fees for a specific credit count, say 11 credits, plug in this number into the function. This stepwise substitution helps:

• Substitute \(x = 11\) in our formula: \(f(11) = 192 \times 11 + 275\)
• Calculating \(192 \times 11\) gives \(2112\)
• Add the fixed fee: \(2112 + 275 = 2387\)

Thus, the total cost for 11 credits is \(2387\). This calculation underscores how credits directly influence tuition.

Fixed Fees

Fixed fees, like the \(275\) mentioned here, play a pivotal role in tuition computation. These are not affected by how many credits you take. Why is it called "fixed"? Because it remains constant. Whether you enroll for a single credit or a full course load, this amount does not change.That makes planning easier as it simplifies budgeting. Simply knowing the fixed fee helps in estimating overall costs, allowing for precise financial preparation. In our function formula, \(275\) was clearly added as a standalone value, showing its non-dependence on credit counts.