Problem 80

Question

In the year 2009 , Olivia's bank balance is 1000 dollars. In the year 2010 , her balance is 1100 dollars. (a) If her balance is growing exponentially, in what year will it reach 2500 dollars? (b) If her balance is instead growing linearly, in what year will it reach 2500 dollars ?

Step-by-Step Solution

Verified

Answer

Answer: Olivia's bank balance will reach $2500 in the year 2018 if it grows exponentially, and in the year 2024 if it grows linearly.

1Step 1: Part (a) Exponential Growth

In this case, we are given that the balance grows exponentially. To find the year when her balance will reach $2500, we will use the exponential growth formula: $A=P(1+r)^{t}$ We are given the initial amount $P=1000$ and the amount at the end of 1 year, which is $A=1100$. We will use this to find the value of $r$. We will also plug the final amount, $A=2500$, in the formula and then solve for $t$ to determine the year when her balance will be $2500. 1. Finding the growth rate: $1100 = 1000(1+r)^{1}$ $r = 0.1$ 2. Finding the number of years ($t$) to reach $2500: $2500 = 1000(1+0.1)^{t}$ $(1+0.1)^{t} = 2.5$ $t\approx 8.1$ Since we can't have a fraction of a year, we will have to round up to the next whole number, which is $9$. Therefore, in the case of exponential growth, it will take Olivia 9 years to reach a balance of $2500. The starting year is 2009, so the year her balance will reach $2500 in the case of exponential growth is: $2009 + 9 = 2018$

2Step 2: Part (b) Linear Growth

In this case, we are given that the balance grows linearly. To find the year when her balance will reach $2500, we will use the linear growth formula: $A = P + rt$ We are given the initial amount $P=1000$ and the amount at the end of 1 year, which is $A=1100$. We will use this to find the value of $r$. We will also plug the final amount, $A=2500$, in the formula and then solve for $t$ to determine the year when her balance will be $2500. 1. Finding the growth rate: $1100 = 1000 + r(1)$ $r = 100$ 2. Finding the number of years ($t$) to reach $2500: $2500 = 1000 + 100t$ $100t = 1500$ $t=15$ So in the case of linear growth, it will take Olivia 15 years to reach a balance of $2500. The starting year is 2009, so the year her balance will reach $2500 in the case of linear growth is: $2009 + 15 = 2024$ In conclusion, Olivia's bank balance will reach $2500 in the year 2018 if it grows exponentially, and in the year 2024 if it grows linearly.

Key Concepts

Exponential Growth FormulaLinear Growth FormulaCalculating Growth RateProjecting Future Value

Exponential Growth Formula

When a quantity increases over time by a constant percentage, this is known as exponential growth. The formula used to calculate the future value of an exponential growing quantity is
\[ A = P(1 + r)^t \]
where

$ A $ is the amount of money accumulated after $ n $ years, including interest.
$ P $ is the principal amount (the initial amount of money).
$ r $ is the annual growth rate (as a decimal).
$ t $ is the time the money is invested or borrowed for, in years.

To illustrate this formula, let's consider the scenario where Olivia's bank balance grows from $1000 to $1100 in one year. To find the growth rate, you would divide the increase ($100) by the original balance ($1000), giving you $ r = 0.1 $, or a 10% growth rate per year. This rate then helps to project the future balance, assuming the growth continues at the same rate. With exponential growth, values can increase rapidly, as each year the growth is compounded.

Linear Growth Formula

In contrast to exponential growth, linear growth increases by an absolute amount each time period rather than by a percentage. The linear growth formula is given by
\[ A = P + rt \]

$ A $ represents the future value of the investment/loan, including the increase.
$ P $ is the principal amount or initial quantity.
$ r $ is the rate of increase per time period.
$ t $ is the number of time periods that have passed.

For example, if Olivia's bank balance grows by $100 each year, the growth rate is \( r = 100 $ when considering a time period of one year. To find out when the balance would reach \)2500, you would solve for $ t $ in the linear growth equation. This shows that growth is steady and predictable over time when using the linear model, which can be easier to calculate and understand than exponential growth.

Calculating Growth Rate

The growth rate is a crucial variable for both exponential and linear growth models. It represents the speed at which the quantity is increasing over time. Calculating the growth rate correctly is vital to ensure accurate projections for both models.
For an exponential growth rate, you generally use the following steps:

Identify the initial value, $ P $, and the value after one time period, $ A $.
Use the formula $ A = P(1 + r)^t $ to find the growth rate. Since $ t $ is often 1 in these calculations, you simplify to $ A = P(1 + r) $.
Rearrange to solve for $ r $, which will give you the percentage increase per time period.

For linear growth, the process is more straight-forward:

You still start with the initial value, $ P $, and the value after one time period, $ A $.
The linear formula, $ A = P + rt $, when $ t $ is 1, simplifies to $ A = P + r $.
Rearrange to solve for $ r $, which is the absolute increase per time period.

Applying these methods will allow you to calculate the yearly growth rate for both exponential and linear scenarios, as illustrated in Olivia's bank balance example.

Projecting Future Value

Projecting the future value of an investment, savings account, or loan balance is an application of both exponential and linear growth models. To project future value:

Identify the type of growth (exponential or linear).
Use the appropriate formula and insert the known values.
Calculate the growth rate if not provided.
Solve for the future value after the desired number of time periods.

With an accurate growth rate and the proper formula, one can determine the amount of money to be expected in the future. For example, in Olivia's case, using the exponential growth formula and her initial balance and growth rate, we can project that after approximately 9 years her balance will grow to $2500. Under linear growth assumptions, however, it will take her 15 years to reach the same amount. Hence, understanding these concepts not only helps in solving textbook problems but also in making informed financial decisions.

Problem 79

Problem 80

Other exercises in this chapter

Problem 79

(a) Using the viewing window with $-4 \leq x \leq 4$ and $-1 \leq y \leq 8, \operatorname{graph} f(x)=\left(\frac{1}{2}\right)^{x}$ and $g(x)=2^{x}$ on th

View solution

Problem 79

The height $h$ above sea level (in meters) is related to air temperature $t$ (in degrees Celsius), the atmospheric pressure $p$ (in centimeters of mercury

View solution

Problem 80

Look back at Section $4.4,$ where the basic properties of graphs of polynomial functions were discussed. Then review the basic properties of the graph of \(f(

View solution

Problem 81

According to one theory of learning, the number of words per minute $N$ that a person can type after $t$ weeks of practice is given by \(N=c\left(1-e^{-k t}

View solution