Problem 40 - Algebra A Combined Function

Question

Use the formula \(A=P\left(1+\frac{r}{n}\right)^{n t}\) to solve these compound interest problems. Round to the nearest tenth. How long does it take for \(\$ 1000\) to double if it is invested at \(8 \%\) interest compounded monthly?

Step-by-Step Solution

Answer

It takes approximately 8.7 years for $1000 to double at 8% interest compounded monthly.

Step 1: Define the Variables

To start solving the problem, we need to identify and define the variables in the formula. In this problem, we have:- The principal amount, \(P = 1000\).- The final amount, \(A = 2000\), because the amount doubles.- The annual interest rate, \(r = 0.08\) (8% as a decimal).- The number of times interest is compounded per year, \(n = 12\) (monthly).

Step 2: Set Up the Equation

Substitute the known values into the compound interest formula:\[2000 = 1000 \left(1 + \frac{0.08}{12}\right)^{12t}\]Here, \(t\) is what we're trying to find.

Step 3: Simplify the Equation

First, simplify the term inside the parentheses:\(1 + \frac{0.08}{12} = 1 + 0.0066667 = 1.0066667\).Now the equation becomes:\[2000 = 1000 (1.0066667)^{12t}\]

Step 4: Divide Both Sides by the Principal

Divide both sides of the equation by 1000 to isolate the exponential term:\[2 = (1.0066667)^{12t}\]

Step 5: Take the Logarithm of Both Sides

Take the logarithm of both sides to solve for \(t\):\[\log(2) = 12t \cdot \log(1.0066667)\]

Step 6: Solve for Time, \(t\)

Rearrange the equation to solve for \(t\):\[t = \frac{\log(2)}{12 \cdot \log(1.0066667)}\]Calculate:\(\log(2) \approx 0.3010\) and \(\log(1.0066667) \approx 0.00289\).Substitute back into the equation:\[t = \frac{0.3010}{12 \times 0.00289} \approx 8.7\]

Step 7: Round the Result

The answer from Step 6 is approximately 8.7 years. Since the problem asks to round to the nearest tenth, we find that the time it takes is approximately 8.7 years.

Key Concepts

Exponential Growth

Exponential growth describes how quantities grow rapidly over time, following a specific pattern. In the realm of finance, compound interest is a classic example of exponential growth. Here, the amount of money you invest grows exponentially as interest is calculated on both the initial principal and the accumulated interest from previous periods.

This differs from linear growth, where the amount increases by a fixed sum and not a percentage. In exponential growth:

• The base amount increases by a consistent percentage.
• The increase is added back to the initial quantity, leading to larger growth over time.

For example, when compounding monthly as in our problem, the interest grows based on what has been previously accumulated and leads to the dramatic increase in value. This is especially useful in scenarios like savings accounts, where leaving your money to mature over time results in significant growth.

Interest Rate

Interest rate is essentially the cost of borrowing money or the reward for saving it. In the context of compound interest, it is crucial as it directly influences how fast your investment grows.

Interest rates can be described in two ways:

• Nominal Interest Rate: This is the stated rate before taking compounding into account.
• Effective Interest Rate: This considers compound interest and reflects the actual growth factor over the period.

In our example, the nominal interest rate of 8% is used. Since compounding occurs monthly, this rate is adjusted by dividing by 12 to find the monthly growth factor. Thus, the interest rate directly shapes how fast exponential growth can occur, and understanding it allows you to compare investment opportunities and potential returns accurately.

Logarithmic Functions

Logarithmic functions are used to solve for an unknown exponent in equations like those involving exponential growth. In the context of compound interest problems, taking the logarithm allows us to isolate the variable we are interested in, typically the time it takes for an investment to grow to a certain amount.

In the problem at hand, we needed to find out how long it would take for money to double under a certain interest rate. By expressing both sides of the equation in logarithmic terms:

• We transform the exponential function into a linear relationship.
• This aids in rearranging the equation so we can solve for the unknown time.

Logarithms simplify multiplications to additions and help handle the complexities of exponential equations. Knowing how to use them effectively is key in solving compound interest problems, where time and growth rates need careful calculation and understanding.