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Chapter 3: Problem 22

Suppose 8000 is deposited in a bank account paying \(7 \%\) interest per year, compounded 12 times per year. How much will be in the bank account at the end of 100 years?

Short Answer

Expert verified
The final amount in the bank account after 100 years is approximately $26,317,994.71, calculated using the compound interest formula with variables \(P=8,000\), \(r=0.07\), \(n=12\), and \(t=100\).

Step by step solution

01

Identify all the variables from the problem

In our problem, we have the following variables: - Initial deposit \(P=8000\) - Annual interest rate \(r=7\% = 0.07\) - Number of times the interest is compounded per year \(n=12\) - Number of years \(t=100\)
02

Substitute the variables into the compound interest formula

Now we will substitute these values into the formula: \[A=8000(1+\frac{0.07}{12})^{12(100)}\]
03

Calculate the inner terms of the formula

First, we need to calculate the terms inside the parentheses: \(1+\frac{0.07}{12} = 1+\frac{7}{100 \times 12}\) Next, evaluate this expression: \(1+\frac{7}{1200} \approx 1.005833\)
04

Raise the inner terms to the power of 1200

Now we take the result from Step 3 and raise it to the power of \(12\times100\): \((1.005833)^{1200} \approx 3289.749363\)
05

Multiply the initial deposit by the result from Step 4

To find the future value of the investment, multiply the initial deposit by the result from Step 4: \(8000\times3289.749363 \approx 26317994.71\)
06

Give the final answer

The final amount in the bank account after 100 years will be approximately: $26,317,994.71.

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