Problem 64 - Brief Applied Calculus

Question

63-64. GENERAL: College Tuition The following graph shows the average annual college tuition costs (tuition and fees) for a year at a private nonprofit or public fouryear college. The data are given for five-year intervals. The tuition for a public college is approximated by the function \(f(x)=400 x^{2}+500 x+2700\), where \(x\) is the number of five-year intervals since the academic year 1995-96 (so the years in the graph are numbered \(x=0\) through \(x=3)\). a. Use this function to predict tuition in the academic year 2020-21. [Hint: What \(x\) -value corresponds to that year?] b. Find the derivative of this function for the \(x\) -value that you used in part (a) and interpret it as a rate of change in the proper units. c. From your answer to part (b), estimate how rapidly tuition will be increasing per year in 2020-21.

Step-by-Step Solution

Answer

Predicted tuition for 2020-21 is $15,200. The rate of tuition increase is $900 per year.

Step 1: Determine the Value of x for the Year 2020-21

Since the intervals start from the academic year 1995-96, each interval represents five years. - For the academic year 1995-96, \(x = 0\).- Therefore, 2020-21 is approximately 25 years after 1995 (2020 - 1995 = 25).- Thus, \(x = 25 \div 5 = 5\) five-year intervals.

Step 2: Predict Tuition in 2020-21 Using Function

Substitute \(x = 5\) into the function \(f(x)=400x^2 + 500x + 2700\).\[f(5) = 400(5)^2 + 500(5) + 2700\]Simplify the expression:\[f(5) = 400 \times 25 + 500 \times 5 + 2700\]\[f(5) = 10000 + 2500 + 2700 = 15200\]Thus, the predicted tuition is $15,200.

Step 3: Find the Derivative of the Function

Differentiate the function \(f(x) = 400x^2 + 500x + 2700\) with respect to \(x\).\[f'(x) = \frac{d}{dx}(400x^2 + 500x + 2700) = 800x + 500\]

Step 4: Use Derivative to Estimate Rate of Change in 2020-21

Substitute \(x = 5\) into the derivative:\[f'(5) = 800(5) + 500\]\[f'(5) = 4000 + 500 = 4500\]This result means that the tuition is increasing at a rate of $4500 per five-year interval.

Step 5: Convert Rate to Increase Per Year

To find the annual increase, divide the increase per interval by the number of years in each interval:\[\text{Annual Increase} = \frac{4500}{5} = 900\]Thus, the tuition is estimated to increase by $900 per year in 2020-21.

Key Concepts

College Tuition Costs Prediction

Predicting college tuition costs is crucial for both students and policy makers. This prediction allows better financial planning and decision-making. In the given exercise, tuition prediction is tackled using a quadratic function, which represents a model of tuition changes over five-year intervals. By substituting an appropriate interval value into the function, future tuition costs can be estimated. In this case, for the year 2020-21, we first determine the number of intervals since the base year (1995-96), which calculates to 5 intervals. By plugging this into the function:

• \( f(x)=400x^2 + 500x + 2700 \)
• For \( x = 5 \), the predicted tuition is $15,200 for a public college.

This use of functions for predictions highlights how mathematical models can make planning more efficient and informative.

Derivatives and Rates of Change

In calculus, derivatives are incredibly useful for understanding how a function behaves over time. They represent the rate of change of a function in relation to its variables. Here, differentiating the tuition cost function tells us how quickly tuition fees are increasing as time progresses. The derivative of the function:

• \( f'(x) = 800x + 500 \)

provides valuable insight into the rate of change of tuition. By calculating the derivative at a specific interval value (for 2020-21, with \( x = 5 \)), the function's rate of increase is determined as $4,500 per five-year interval. This indicates that not only can we predict the amount of tuition, but we can also understand how rapidly it is increasing over time.

Mathematical Modeling in Education

Mathematical modeling plays a significant role in education, particularly in understanding trends and making future projections. By using models, we can simulate real-world phenomena and reach meaningful conclusions. In the case of college tuition, the quadratic model captures the historical increase and projects it into the future. This approach is not unique to tuition but can be used across various educational financial scenarios.

• Models help identify patterns and predict variables that change over time.
• The reliability of these models depends on the accuracy of data and assumptions made.

Incorporating derivatives enhances these models by showing the dynamics of change, further assisting administrators and students in planning for costs. Thus, this mathematical approach offers a robust toolset for tackling educational financial planning challenges.