91Ó°ÊÓ

The yearly membership for a professional organization is \(\$ 250\) per year for the current year and increases by \(\$ 25\) per year. If a person joins for \(x\) consecutive years, the average cost per year \(\overline{C_{1}}(x)\) (in \$) is given by $$\overline{C_{1}}(x)=\frac{475+25 x}{2}$$ a. Find the average cost per year if a person joins for \(5 \mathrm{yr}, 10 \mathrm{yr},\) and \(15 \mathrm{yr}\) b. The professional organization also offers a one-time fee of \(\$ 2000\) for a lifetime membership. If a person purchases a lifetime membership, write an average cost function representing the average cost per year \(\overline{C_{2}}(x)\) (in \$) for \(x\) years of membership. c. If a person purchases a lifetime membership, compute the average cost per year for \(5 \mathrm{yr}, 10 \mathrm{yr},\) and \(15 \mathrm{yr}\). d. Interpret the meaning of the horizontal asymptote for the graph of \(y=\overline{C_{2}}(x)\)

Step by step solution

01

Calculate average cost for 5 years

Given \(\overline{C_{1}}(x)=\frac{475+25x}{2}\), substituting \(x = 5\): \(\overline{C_{1}}(5) = \frac{475 + 25 \times 5}{2} = \frac{475 + 125}{2} = \frac{600}{2} = 300 \$ \)

02

Calculate average cost for 10 years

Substituting \(x = 10\) into \(\overline{C_{1}}(x)\): \(\overline{C_{1}}(10) = \frac{475 + 25 \times 10}{2} = \frac{475 + 250}{2} = \frac{725}{2} = 362.5 \$ \)

03

Calculate average cost for 15 years

Substituting \(x = 15\) into \(\overline{C_{1}}(x)\): \(\overline{C_{1}}(15) = \frac{475 + 25 \times 15}{2} = \frac{475 + 375}{2} = \frac{850}{2} = 425 \$ \)

04

Define the average cost function for lifetime membership

The one-time fee is \$ 2000. The average cost per year for \(x\) years of membership is given by: \(\overline{C_{2}}(x) = \frac{2000}{x} \)

05

Calculate average cost for 5 years of lifetime membership

Substituting \(x = 5\) into \(\overline{C_{2}}(x)\): \(\overline{C_{2}}(5) = \frac{2000}{5} = 400 \$ \)

06

Calculate average cost for 10 years of lifetime membership

Substituting \(x = 10\) into \(\overline{C_{2}}(x)\): \(\overline{C_{2}}(10) = \frac{2000}{10} = 200 \$ \)

07

Calculate average cost for 15 years of lifetime membership

Substituting \(x = 15\) into \(\overline{C_{2}}(x)\): \(\overline{C_{2}}(15) = \frac{2000}{15} \approx 133.33 \$ \)

08

Interpret horizontal asymptote for \(\overline{C_{2}}(x)\)

As \(x\) goes to infinity, the average cost per year \(\overline{C_{2}}(x)\) approaches 0, meaning that for a very large number of years, the average cost per year for lifetime membership becomes negligible.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Yearly Membership Cost

Understanding the yearly membership cost is crucial. For the current year, the cost is \(250. However, it increases by \)25 each year. This stepwise increase means that every year you stay a member, the cost goes up. This increase can make a significant difference over time. Knowing this helps you plan your finances better.

Average Cost Per Year

To find the average cost per year for a specified number of years, we use a given formula: \ \ \ \( \overline{C_{1}}(x)=\frac{475+25x}{2} \) \ \ \ This formula helps us compute the yearly average over any number of years. For example, if you want to find the average cost for 5 years, you substitute \( x = 5 \) into the formula: \ \ \ \ \ \( \overline{C_{1}}(5) = \frac{475 + 25 \times 5}{2} = 300 \$ \) \ \ \ Similarly, for 10 years and 15 years, the average costs are 362.5\$ and 425\$, respectively. This helps understand how your costs will change over time.

Lifetime Membership Cost

A lifetime membership offers another option. It costs a one-time fee of $2000. Over time, you can consider this fee as an average yearly cost by using the formula: \ \ \ \( \overline{C_{2}}(x) = \frac{2000}{x} \) \ \ \ For example, for 5 years, the average would be: \ \ \ \ \ \( \overline{C_{2}}(5) = \frac{2000}{5} = 400 \$ \) \ \ \ For 10 years and 15 years, the averages come out to be 200\$ and approximately 133.33\$, respectively. This helps you realize the value of a lifetime membership over an extended period.

Horizontal Asymptote

A horizontal asymptote gives an insight into the long-term behavior of a function. For the lifetime membership average cost function \( \overline{C_{2}}(x) = \frac{2000}{x} \), as \( x \) (number of years) goes to infinity, the function value approaches 0. This means that the average cost per year becomes negligible over a very long period. Essentially, the longer you remain a member, the more cost-effective your lifetime membership becomes, approaching almost zero cost in the long run.