Problem 50 - Algebra 2

Question

Write an equation. Then solve the equation without graphing. A parent raises a child's allowance by 20\(\%\) each year. If the allowance is \(\$ 8\) now, when will it reach \(\$ 20 ?\)

Step-by-Step Solution

Answer

The value of \( t \) will be the number of years it will take for the child's allowance to reach $20.

Step 1: Set Up the Equation

The allowance grows at a rate of 20% each year. This growth can be represented by the equation \( A = P(1 + r)^t \), where \( A \) is the final amount, \( P \) is the initial amount, \( r \) is the rate of increase, and \( t \) is time. Here, the initial amount \( P = \$8 \), the rate of increase \( r = 20\% = 0.2 \), and the final amount \( A = \$20 \). Plug these values into the equation to get, \( 20 = 8(1 + 0.2)^t \).

Step 2: Solve the Equation

Solving the equation \( 20 = 8(1.2)^t \) for \( t \), divide by 8 from both sides to get the equation as, \( 2.5 = (1.2)^t \). Taking natural logarithm (ln) on both sides will give, \( \ln(2.5) = t \ln(1.2) \). Thus, \( t = \frac{\ln(2.5)}{\ln(1.2)} \).

Step 3: Evaluate Time

Use a calculator to evaluate the expression \( t = \frac{\ln(2.5)}{\ln(1.2)} \), to find the value of \( t \).

Key Concepts

Compound Interest

Compound interest is a key concept in understanding exponential growth. Imagine your money growing at a specific rate each year, where each year's growth is based on the previous year's total. This is what's happening with compound interest.
In the allowance problem, the allowance increases yearly by 20%, similar to how money would grow in a compound interest account. The formula for compound interest is:

• \( A = P(1 + r)^t \)

Here, \( A \) represents the amount after a certain time, \( P \) is the initial amount (\\(8 in this problem), \( r \) is the rate of increase (here 0.2 or 20%), and \( t \) is time in years. This formula helps to calculate how much the allowance will be in the future as it grows exponentially.
In the exercise, this equation was set up and solved to find out how long it takes for the allowance to reach \\)20.

Percentage Increase

Percentage increase is an essential part of understanding how an amount grows over time. In simple terms, a percentage increase means that instead of adding a fixed amount each year, you add a proportion of the initial amount.
For an allowance increasing by 20% each year, you add 20% of the allowance from the previous year. For example, if the allowance is initially \\(8, the first year's increase is \\)1.60 (20% of \\(8), raising the allowance to \\)9.60.
This continues year after year, with each year's increase being calculated as 20% of the latest allowance amount, resulting in a faster growth than simple addition would provide. This compounding nature is why percentage increases are powerful for exponential growth.

Logarithms

Logarithms are very important when solving exponential problems, especially when trying to find the unknown time in compound growth. They help "flatten" exponential growth to linear terms, making calculations easier.
When solving for the time \( t \) in an exponential growth problem, you often end up needing to "undo" the exponent. This is where logarithms come in.
In our exercise, after rearranging the formula to \( 2.5 = (1.2)^t \), the next step used logarithms:

• \( \ln(2.5) = t \ln(1.2) \)

Here, taking the natural logarithm, represented as \( \ln \, \), helped solve for \( t \). By dividing \( \ln(2.5) \) by \( \ln(1.2) \), you could find how many years it would take for the allowance to reach \$20.

Time Calculation

Time calculation in exponential growth problems often involves isolating the time variable after using logarithms. After establishing the relationship and setting up the equation, you'll need to rearrange so "time" is the subject.
In the given exercise, this was handled by first dividing \( 2.5 = (1.2)^t \) to segregate \( t \) on one side. Then logarithms were applied, allowing us to express \( t \) as:

• \( t = \frac{\ln(2.5)}{\ln(1.2)} \)

The calculator is then used to evaluate \( t \), completing the time calculation. Finding \( t \) provides the number of years needed for the allowance to reach \$20. Understanding time calculation is crucial to solving real-world exponential problems as it determines how quickly growth targets are met.