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Chapter 4: Problem 6

How long will it take a bank deposit to triple in value if interest is compounded continuously at a constant rate of \(5 \frac{1}{4}\) percent per annum?

Short Answer

Expert verified
Answer: It will take approximately 26.070 years for the bank deposit to triple in value.

Step by step solution

01

Understand the formula for continuous compounding

The formula for continuous compounding is given by \(A(t) = P * e^{rt}\), where A(t) is the amount after time t, P is the principal (initial) amount, r is the interest rate, and t is the time (in years).
02

Determine the initial amount, final amount, and interest rate

In this exercise, we are trying to find the time it takes for the money to triple. Therefore, we can say that the initial amount (P) is 1 unit, and the final amount (A) will be 3 units, as it needs to triple. The interest rate (r) is given as \(5 \frac{1}{4}\) percent or \(\frac{21}{4}\) percent, but we need to convert it into a decimal value. To do this, divide the percentage rate by 100, so \(\frac{21}{4} * \frac{1}{100} = \frac{21}{400}\).
03

Set up the equation

Using the formula for continuous compounding and the values we found in Step 2, set up the equation as follows: \(3 = 1 * e^{\frac{21}{400}t}\)
04

Solve for time (t)

To solve for t, we will first divide both sides of the equation by 1 (which doesn't change anything) and then take the natural logarithm (ln) of both sides: \(3 = e^{\frac{21}{400}t}\) \(\ln{3} = \ln{e^{\frac{21}{400}t}}\) Now, use the property of logarithms where \(\ln{e^x} = x\): \(\ln{3} = \frac{21}{400}t\) Next, isolate t by dividing both sides by \(\frac{21}{400}\): \(t = \frac{\ln{3}}{\tfrac{21}{400}}\) \( t = \frac{400\ln{3}}{21}\) Using a calculator to evaluate this expression: \(t = \approx 26.070\) years
05

Interpret the result

The answer is approximately 26.070 years. Therefore, it will take about 26.070 years for the bank deposit to triple in value if the interest is compounded continuously at a constant rate of \(5 \frac{1}{4}\) percent per annum.

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Solve \(\left(\frac{\sin 2 x}{y}+x\right) d x+\left(y-\frac{\sin ^{2} x}{y^{2}}\right) d y=0\).

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