4.5.2: Exercise 4.5

MAKING CONNECTIONS TO THE COLLABORATION

(1) Which of the following was one of the main mathematical ideas of the collaboration?

(i) Algebra gives us a way to write general rules that can be used in multiple situations.

(ii) Compound interest is an important concept in savings and investment.

(iii) A linear model never uses percentages but an exponential model does.

(iv) Annual compounding is done once a year.

DEVELOPING SKILLS AND UNDERSTANDING

(2) Write a statement about the rate of change for each model below:

(3) Recall the simple interest formula used earlier in this module:

A = P + P ⋅ r ⋅ t

Where A is the Amount (principal and interest in the account, also called the balance), P is the Principal, r is the annual interest rate (quoted as a percent, but used as a decimal), and t is the time in years.

(a) Complete the table below with the amounts you would have if you invested $1,000 at 5% interest with simple interest and with interest compounded annually.

(b) Is simple interest a linear model? How do you know?

(4) Which investment strategy will have a higher balance after 10 years: investing $500 at 10% APR or $1,000 at 2% APR? Both investments have annual compounding.

(a) After 10 years, the $500 at 10% APR would grow to $_____________ .

(b) After 10 years, the $1,000 at 2% APR would grow to $_____________ .

(c) Which investment strategy would have a higher balance after 10 years?

i. $500 at 10%

ii. $1000 at 2%

MUTUAL FUNDS

CDs are a very safe investment because they are usually insured by the U.S. government. (There are some CDs that are not insured so it is important to always check!) Because they are so safe, CDs earn low rates of interest. The amount earned on an investment is often called the return. In general, investments with higher risk also have the potential for higher rates of return. For example, if you invest in a stock, which is like buying a piece of a company, you might earn far more than you would with a CD. However, you also run the risk of losing all of your money.

Mutual funds are another type of investment used by many people. The U.S. Security and Exchange Commission defines a mutual fund as follows:¹⁴

A mutual fund is a company that brings together money from many people and invests it in stocks, bonds, or other assets. The combined holdings of stocks, bonds, or other assets the fund owns are known as its portfolio. Each investor in the fund owns shares, which represent a part of these holdings.

Mutual funds are an attractive investment to many people for several reasons.

• You can invest small amounts of money at a time.
• The fund is managed by a company that does all the work of researching and choosing specific investments.
• In general, there is less risk than owning a single stock because the money is spread across many investments.
• In general, there are higher rates of returns than those available in insured investments like CDs and savings accounts.

However, it is important to understand that mutual funds do still have risks. It is possible to lose some or all of your investment. Also, funds charge fees to pay for the management. Sometimes these fees can be very high.

Selecting investments is an important decision and should be researched carefully. Through this course, you will be introduced to a few basic concepts related to investment, but you should not make any decisions based only on the information presented here.

(5) One of the most important concepts in investing is to take advantage of the power of compounding over time. The following example will help you explore this idea.

Lorenzo and Michael each decide to invest in a mutual fund to save for retirement. They each choose a mutual fund that has had an average return of 5.6% over the last decade. There is no guarantee that the mutual fund will continue to earn this same rate, but it can be used as an estimate of future returns. Use the information given below to estimate how much each man will have when he retires. Round to the nearest dollar. (Hint: The amount earned is how much they have without the initial investment.)

• Lorenzo invests $2,000 when he is 25 years old.
• Michael invests $5,000 when he is 45 years old.
• Both men plan to retire at age 65.

MAKING CONNECTIONS ACROSS THE COURSE

As with linear models, exponential models can increase, which is called exponential growth, or decrease, which is called exponential decay. You will see an example of exponential decay in Collaboration 4.6.

Compound interest is one application of exponential growth. There are many others. The formula that you found for annual compound interest can be generalized to all exponential growth models:

\[New\;Value = Starting\;Value\times (1 + rate)^{variable\;power}\nonumber \]

Populations often grow exponentially. Question 7 examines how you can use this to make predictions about the world population.

(6) In Collaboration 1.2, you looked at historical data of the world’s population. Scientists use models to project population in the future. While these models have limitations, they help leaders plan ahead for the resources people will need. The U.S. Census Bureau estimated the 2023 world population to be about 7.94 billion people. They estimated the world population growth rate in 2023 to be about 0.94%.¹⁵

(a) This is an exponential situation because the population is increasing each year by a percentage of the previous year. Complete the table below based on the growth rate of 0.94%. Some of the entries are done for you. Round the populations to the nearest tenth of a billion. (Hint: 5% written as a decimal is 0.05. 4.5% written as a decimal is 0.045. How would 0.94% be written as a decimal?)

(b) Write an exponential equation for the world population growth after 2023. Let P = the projected population in billions and t = the number of years after 2023. (Hint: Think about how you calculated the entries in the table. If you started with 7.94 billion each time, how could you calculate each population? Think back to your work in Collaboration 4.5.)

(c) Use your model to predict the population in 2033.

(7) The Center for Disease Control (CDC) published the following information about emergency department visits for respiratory illness from a meta-analysis of data collected in 2014.¹⁶ Respiratory disease is any illness that affects a person’s ability to breathe and use oxygen.

In 2014, approximately 400,000 emergency department visits for respiratory illness. Of these, 7% were for pneumonia, 15% for asthma, 17% for bronchitis, 40% for acute pharyngitis, and 21% for other respiratory diseases, including influenza.

(a) Based on this information, how many people were diagnosed with pneumonia in emergency departments in 2014?

(b) Which of the following pie charts accurately represents the data?

_____________________________________________

¹⁴ http://www.sec.gov/investor/tools/mfcc/mutual-fund-help.htm

¹⁵ https://www.census.gov/newsroom/press-releases/2022/new-years-day-population.html

¹⁶ https://www.cdc.gov/nchs/data/nhsr/nhsr151-508.pdf