Problem 74 - Contemporary Precalculus

Question

If inflation runs at a steady \(3 \%\) per year, then the amount a dollar is worth decreases by \(3 \%\) each year. (a) Write the rule of a function that gives the value of a dollar in year \(x .\) (b) How much will the dollar be worth in 5 years? In 10 years? (c) How many years will it take before today's dollar is worth only a dime?

Step-by-Step Solution

Answer

Answer: After 5 years, the dollar will be worth approximately $0.86. After 10 years, the dollar will be worth about $0.74. It will take approximately 33 years for the dollar to be worth only a dime.

Step 1: (1) Understand the concept of Exponential decay and find the function for dollar value)

Exponential decay means that something decreases by a certain percentage every year. In our case, the dollar's value will decrease by \(3 \%\) (0.03) each year. To create a function to get the dollar value, we can use the following general formula for decay: $$ A(x) = A_0 * (1 - r)^x $$, where A(x) is the value of the dollar in year x, A_0 is the initial value, r is the rate at which the dollar is decreasing (i.e., \(3 \%\)), and x is the number of years. Now, we are ready to find the function that represents the value of a dollar in year x.

Step 2: (2) Create the function to represent the value of a dollar)

We are given the initial value (\(A_0\)) of a dollar as \(1\), and the rate (\(r\)) is \(3 \%\) or \(0.03\). We can insert these values into the exponential decay formula to obtain the function for the value of a dollar over time: $$ A(x) = 1 * (1 - 0.03)^x $$. (a) So, the rule of a function that gives the value of a dollar in year x is: $$ A(x) = (0.97)^x $$.

Step 3: (3) Calculate the value of the dollar in 5 and 10 years)

Now that we have our function, we can use it to determine the value of the dollar in 5 and 10 years. To do so, we will insert x=5 for 5 years and x=10 for 10 years in the function: (b) 5 years: $$ A(5) = (0.97)^5 = 0.858734 $$. In 5 years, the dollar will be worth approximately \(0.86\). 10 years: $$ A(10) = 0.97^{10} = 0.7374 $$. In 10 years, the dollar will be worth about \(0.74\).

Step 4: (4) Find the number of years for the dollar to be worth only a dime)

To find when the dollar is worth only a dime (\(0.1\)), we need to solve for x in the function: $$ A(x) = 0.1 $$ $$ 0.1 = (0.97)^x $$ (c) To solve for x, we can use logarithms: $$ x \log(0.97) = \log(0.1) $$ We’ll divide by \(\log(0.97)\) to isolate x: $$ x = \frac{\log(0.1)}{\log(0.97)} = 32.95 $$ Thus, it will take approximately \(33\) years for today's dollar to be worth only a dime.

Key Concepts

Exponential Decay Formula

When we talk about exponential decay, we're describing a process whereby a quantity decreases at a rate proportional to its current value.
In the context of finance and economics, this is often used to model the diminishing value of money over time due to inflation. The concept also finds application in other fields like radioactive decay in physics or population decline in biology.
The exponential decay formula is generally expressed as:
\[ A(x) = A_0 \times (1 - r)^x \],
where \( A(x) \) is the value of the quantity at time \( x \), \( A_0 \) is the initial value, \( r \) is the decay rate, and \( x \) is the time elapsed.
For instance, if you have \( 1 \) dollar initially and it depreciates by \( 3\text{%} \) per year due to inflation, the decay rate \( r \) would be \( 0.03 \), and the formula to find out its worth in any subsequent year would become \( A(x) = (1 - 0.03)^x \).
Applying this to our exercise, the rule of the function that gives the value of a dollar in year \( x \) is expressed as \( A(x) = (0.97)^x \).
This concise representation allows us to easily calculate the future value of a dollar, given its annual decay.

Future Value Calculation

Understanding the future value calculation is essential for anyone looking to grasp how investments grow over time or how value changes with inflation.
In our discussed problem, we've used the exponential decay formula to compute the decrease in the value of a dollar over a period of time. However, this method can also be adapted for understanding the future value of investments through a similar technique called 'compound interest.'
In the compound interest formula, the future value is calculated using the formula:
\[ A = P \times (1 + r/n)^{nt} \],
where \( P \) is the principal amount, \( r \) is the annual interest rate, \( n \) is the number of times interest is compounded per year, and \( t \) is the number of years the money is invested or borrowed for.
Turning back to our textbook scenario where we're dealing with exponential decay rather than growth, we calculated:

• Future value of a dollar in 5 years: \( 0.97^5 \) which equals roughly \( 0.86 \).
• Future value of a dollar in 10 years: \( 0.97^{10} \) resulting in approximately \( 0.74 \).

These calculations forecast the reduced purchasing power of money, an important concept for personal finance and investment planning.

Logarithmic Equations

Logarithmic equations are indispensable when it comes to solving problems that involve exponential functions. In the realm of mathematics, a logarithm answers the question: 'To what exponent must we raise a certain number (the base) to obtain another number?'.
In our exercise, we use a logarithmic equation to find out how many years it will take for a dollar to decay to the value of a dime in the face of constant inflation. The form of a logarithmic equation derived from our exponential decay formula is:
\[ x = \frac{\text{log}(A(x))}{\text{log}(1 - r)} \].
Given that we want the future value \( A(x) \) to be \( 0.1 \) (a dime), and using the base \( 0.97 \) for the decay (assuming a constant inflation rate of \( 3\text{%} \)), we rearrange our decay formula to solve for \( x \) using logarithms:

• Logarithmic equation: \( \text{log}(0.1) = x \times \text{log}(0.97) \)
• The solution for \( x \): \( x = \frac{\text{log}(0.1)}{\text{log}(0.97)} \)

This manipulation yields a numerical value of about \( 33 \) years, which means that after this period, a dollar initially would have decayed exponentially to be worth a mere ten cents. Understanding logarithmic equations is, therefore, a powerful tool for demystifying the kinetics of exponential decay and growth processes.