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The yearly cost of tuition (in-state) and required fees for attending a public two-year college full time can be estimated by the linear function \(f(x)=64 x+2083,\) where \(x\) is the number of years after 2000 and \(f(x)\) is the total cost. (Source: The College Board) a. Use this function to approximate the yearly cost of attending a two-year college in the year \(2016 .\) [Hint: Find \(f(16) .]\) b. Use the given function to predict in what year the yearly cost of tuition and required fees will exceed 3200 dollars. I Hint: Let \(f(x)=3200,\) solve for \(x,\) then round your solution up to the next whole year. \(]\) c. Use this function to approximate the yearly cost of attending a two-year college in the present year. If you attend a two-year college, is this amount greater than or less than the amount that is currently charged by the college you attend?

Step by step solution

01

Identify the year for calculation

First, determine the value of \(x\) for the given year 2016. This is calculated as \(x = 2016 - 2000 = 16\).

02

Calculate yearly cost for 2016

Use the equation \(f(x) = 64x + 2083\) to find \(f(16)\).\[f(16) = 64 \times 16 + 2083\]Calculate this to get the result.\[f(16) = 1024 + 2083 = 3107\] dollars.

03

Solve for when cost exceeds $3200

Set \(f(x) = 3200\) and solve for \(x\).\[3200 = 64x + 2083\]Subtract 2083 from both sides:\[3200 - 2083 = 64x\]\[1117 = 64x\]Divide both sides by 64:\[x = \frac{1117}{64} \approx 17.453125\]Round up to the next whole year, \(x = 18\).

04

Determine the year when cost exceeds $3200

Since \(x = 18\), the year is \(2000 + 18 = 2018\). The cost will exceed $3200 in 2018.

05

Calculate cost for current year

Assuming the present year is 2023, calculate \(x = 2023 - 2000 = 23\).Use the function:\[f(23) = 64 \times 23 + 2083\]\[f(23) = 1472 + 2083 = 3555\] dollars.

06

Verification for personal reference

Compare the calculated cost \($3555\) for the present year with the actual tuition cost of your college. This determines whether the value is higher or lower than what you currently pay.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Tuition Costs

Tuition costs refer to the amount of money required to attend courses at a college or university. For a two-year college, these costs often cover classroom learning, lab sessions, and sometimes additional required fees. Tuition fees can vary significantly depending on the institution, location, and resources provided. Understanding the trend of tuition costs over the years is important for budgeting and planning.
In our exercise, we have a linear function that estimates the tuition costs at a two-year college. The function is useful because it simplifies the calculation of tuition costs for any given year, helping students and parents predict future expenses.

Exploring Two-Year Colleges

Two-year colleges, often referred to as community colleges, provide an affordable route to higher education. They offer associate degrees and certificates in various fields, preparing students either for entry-level jobs or for transfer to four-year institutions.
These colleges are particularly appealing due to lower tuition costs compared to four-year universities. Studying at a two-year college allows students to complete foundational courses while saving money and potentially reducing student loan burdens. Our example exercise estimates these tuition costs using a linear function tailored for understanding expenses over time.

Function Approximation in Action

Function approximation is a mathematical method used for estimating complex data with simple functions. In this exercise, we use a linear function to approximate tuition costs. This type of function is a straight line, characterized by its slope and y-intercept.
The slope, 64 in our exercise, represents the rate at which tuition costs increase each year. The y-intercept, 2083, is the estimated base tuition cost in the year 2000. By inputting the number of years since 2000, students can use the function to estimate yearly costs. Linear functions are particularly helpful as they provide direct insights into trends and can be used for making economic decisions.

Predicting Future Tuition Costs

Cost prediction is a powerful tool for planning educational finances. In our case, once we have a linear function, predicting the tuition cost in any future year is straightforward.
Understanding how to solve for when costs exceed a certain threshold, like $3200, is crucial. We solve the function for the year when the tuition exceeds this amount, informing students of potential future financial planning needs. This type of prediction can help students determine when they might need additional financial aid or scholarships to support their education. By understanding this predictive process, students gain the ability to better manage their educational finances.