Problem 33 - Algebra 2 and Trigonometry

Question

The formula \(A=P(1+r)^{t}\) expresses the amount \(A\) to which \(P\) dollars will increase if invested for \(t\) years at a rate of \(r\) per year. What is the minimum number of years that \(\$ 1\) must be in invested at 5\(\%\) to increase to \(\$ 2 ?\) (Use a calculator to try possible values of \(t.\))

Step-by-Step Solution

Answer

The answer is 15 years.

Step 1: Identify Variables

First, let's identify the given variables from the formula \(A=P(1+r)^{t}\). Here, the final amount \(A\) is 2, the initial amount \(P\) is 1, and the rate \(r\) given in percentage must be converted to a decimal. Therefore, \(r = 0.05\). We need to find the time \(t\).

Step 2: Set Up the Equation

Substitute the known values into the equation: \(2 = 1(1+0.05)^{t}\). Simplify the equation to get \(2 = (1.05)^{t}\).

Step 3: Solve by Testing Values

We need to find the minimum \(t\) such that \((1.05)^t=2\). Start testing values: - For \(t=10\), \((1.05)^{10} \approx 1.63\)- For \(t=15\), \((1.05)^{15} \approx 2.08\)Since \(t=15\) is just over two and \(t=14\) less than two, test further.

Step 4: Narrow Down to Exact Value

Check around \(t=15\) with more precision:- For \(t=14\), \((1.05)^{14} \approx 1.98\)- For \(t=15\), \((1.05)^{15} \approx 2.08\).Since \(t=15\) just exceeds 2, it confirms \(t=15\) is The smallest integer value that meets the condition.

Key Concepts

Exponential Growth

In the world of finance, exponential growth is a concept that describes the increase in an investment over time, where the amount grows at a consistent rate. This is different from linear growth, where the increase is by the same absolute amount each period. When money is invested at a fixed annual interest rate, as in our exercise, the growth is exponential. It's because the interest earned each year adds to the principal, becoming part of the base that earns interest in subsequent years. To understand this, consider the formula: - The initial amount is multiplied by an increasing factor each period. For a 5% interest rate, the factor is 1.05 each year. - Over time, this leads to compounding; the investment doesn't grow by 5% of the original amount each year, but by 5% of its own increasing total. Exponential growth results in a curve that becomes steeper over time, illustrating faster increases. This concept is powerful in finance, showing how small, consistent investments can lead to substantial growth over the long run.

Investment Calculations

Investment calculations involve determining the future value of a financial asset based on various inputs such as the initial amount invested, the interest rate, and the duration of the investment. These calculations help investors understand how much they can expect to earn. In our specific case, we're using the formula: \[ A = P(1 + r)^t \]where:

• \( A \) is the final amount after \( t \) years.
• \( P \) is the initial principal amount.
• \( r \) is the annual interest rate (expressed as a decimal).
• \( t \) is the time in years.

To determine the period required to double an investment, we adjust the formula to solve for \( t \). In this exercise, we started with \( P = 1 \) and aim for \( A = 2 \), with an annual interest rate \( r = 0.05 \). This calculation shows us not only the time it takes for an investment to grow to a certain size but also the impact of different rates and times on the final outcome.

Annual Interest Rate

The annual interest rate is a percentage that indicates how much an investment will grow over the course of a year. It's a critical component in calculating compound interest because it shows the rate at which your money increases each year. In our exercise, we consider a 5% annual interest rate, which is equivalent to multiplying the principal by 1.05 each year. When calculating compound interest, it's essential to convert the percentage to a decimal by dividing by 100, hence 5% becomes 0.05. Understanding how the annual interest rate affects the growth of an investment is vital:

• Higher interest rates will accelerate the growth of your investment.
• Even small changes in the rate can significantly impact the outcome over long periods due to the nature of compounding.
• Predicting how much interest you will earn helps in making informed financial decisions.

Analyzing different interest rates helps visualize potential growth paths of investments, allowing for strategic planning and better management of financial goals.