WER - Exponential Growth Models (Lesson)

Exponential Growth Models

Jane buys a rare trading card for $5. The value of the card increases by 40% per year. How much will Jane's trading card be worth in 3 years?

Let's start out by making a table:

Notice that each time we calculated the new value, we used the previous year's value and then found 40% of that new value and added those together. In other words: old value + 40% of old value = new value

You'll notice that the values do not have a constant difference. They are growing by a ratio! If we want to determine what that rate is, we can divide each new value by the year's previous value:

13.72/9.80 = 1.4

9.8/7 = 1.4

7/5 = 1.4

So why is the value of the card growing by a rate of 1.4? Well, we know that we are increasing the value by 40% each year or 0.4 and the card retains its value from the previous year which means that each time we are multiplying by one whole plus the 0.4 or 1.4.

Because the function is growing by a constant ratio, we know that it must be an exponential function. In fact, there is a general formula for an exponential growth model:

Let's check out the equation and graph of the growth of Jane's playing card:

Try to answer these questions:

1. What is the y-intercept of the graph? What does it represent?
2. What is the growth factor 1.4, written as a percent? 
3. Use the graph to estimate the value of the card in 2.5 years. 
4. What is the domain in the context of the problem? 

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Watch this video to explore a few more exponential growth relationships:

Let's see if you've got it:

1. Jill deposits $300 into an account that grows by 15% each year. Write an exponential growth model to describe the balance of Jill's account. 
2. How much money will Jill have after 5 years? 
3. Approximately when will Jill have $1000 in her account? 
4. Lane's parents have offered to double his allowance each month. He starts out earning $1 the first month. Write an exponential growth model to describe how much Lane earns each month.
5. In which month will Lane earn $128?

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Compound Interest

When you deposit money into a bank account, you earn interest on that money. The interest is compounded a certain number of times per year. We use the compound interest formula below to calculate interest earned.

Let's say you deposit $1500 into an account that earns 5% interest compounded monthly. This means that:

P = 1500

r = 5% or 0.05

n = 12 (monthly)

so the function is:          

$A=1500\left(1+\frac{.05}{12}\right)^{12t}$ 

Watch this video to explore compound interest further.

Exponential Growth Models Practice

Try these problems to see if you've got it:

1. Jason deposits $700 in an account that earns 9% interest compounded quarterly. How much will Jason have after 6 years? Solution: $1194.04
2. Harriet deposits $1200 in an account that earns 6% interest compounded monthly. How much will Harriet have after 10 years? Solution: $2183.28

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