Problem 43 - A Graphical Approach to College Algebra

Question

Solve each problem. cost of Tuition The cost of tuition is directly proportional to the number of credits taken. If 11 credits cost \(\$ 720.50,\) find the cost of taking 16 credits. What is the constant of variation?

Step-by-Step Solution

Answer

The cost of 16 credits is $1048; the constant of variation is approximately 65.50.

Step 1: Understand Direct Variation

The problem states that the cost of tuition is directly proportional to the number of credits. This implies that if the number of credits increases, the total cost also increases at a constant rate. Mathematically, this relationship can be expressed as \( C = k \cdot n \), where \( C \) is the cost, \( k \) is the constant of variation, and \( n \) is the number of credits.

Step 2: Use the Given Values to Find the Constant of Variation

We are given that 11 credits cost \( \$720.50 \). Using the equation from Step 1: \( 720.50 = k \cdot 11 \). Solving for \( k \), we get \( k = \frac{720.50}{11} \approx 65.50 \). Thus, the constant of variation \( k \) is approximately \( 65.50 \).

Step 3: Calculate the Cost for 16 Credits

Now that we have found \( k \approx 65.50 \), we can substitute this into the equation \( C = k \cdot n \) to find the cost for 16 credits. Therefore, \( C = 65.50 \cdot 16 = 1048 \). Thus, the cost of 16 credits is \( \$1048 \).

Key Concepts

Constant of Variation

The concept of the constant of variation is fundamental in understanding how two quantities change in relation to each other. In the context of tuition cost calculations, the constant of variation helps determine how the cost changes with each additional credit.
When we say two quantities are directly proportional, it means they increase or decrease at the same rate. The constant rate is known as the "constant of variation." Mathematically, it is represented by the letter *k*.
For example, if the tuition cost varies directly with the number of credits, the equation is given by \[ C = k \cdot n \]where:- *C* is the total cost,- *k* is the constant of variation,- *n* is the number of credits.
In our specific scenario, we found that for 11 credits costing \( \\(720.50 \), the constant of variation *k* is approximately \( 65.50 \). This means for each credit, the cost is \( \\)65.50 \).

Linear Relationships

Linear relationships describe a straight-line connection between two variables. This is exactly what we see when discussing tuition costs based on credits.
A relationship is considered linear if it can be graphed as a straight line. The equation representing this relationship takes the form of \( y = mx + b \), where *m* is the slope. In direct variation, *b*, the y-intercept, is zero, simplifying the equation to \( y = mx \).
Thus, for tuition cost, the equation becomes: \[ C = k \cdot n \]- Here, *C* functions like \( y \),- *k* takes the role of the slope \( m \),- *n* is equivalent to \( x \).
This direct variation implies the cost (\( C \)) increases linearly with the number of credits (\( n \)). Such relationships are easy to predict and understand, especially when calculating costs based on consistent rates.

Tuition Cost Calculations

Calculating tuition costs using direct variation is straightforward once you find the constant of variation. Let's see how this plays out in practice.
Given that our constant of variation \( k \) is \( 65.50 \), we can easily find the total cost for any number of credits. For instance, if you enroll in 16 credits:- Simply use the formula \( C = k \cdot n \).- Plug in the numbers: \( C = 65.50 \cdot 16 \).- Calculate to find \( C = 1048 \).
Therefore, the tuition cost for 16 credits is \( \$1048 \).
Such straightforward calculations can help students make budgetary decisions concerning their credit loads and understand how each additional credit contributes to their total tuition expense.
In summary, knowing the constant of variation allows for swift and accurate calculations, enabling effective financial planning for students.