Question Statement

Suppose that Rs. 50,000 is invested at $6\%$ interest compounded annually. After $t$ years, it grows to the amount $A$ given by the function:

$A(t)=50,000(1.06{)}^t$

a. After what amount of time will Rs. 50,000 grow to Rs. 450,000? b. Find the doubling time.

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Background and Explanation

This problem uses an exponential growth model to calculate compound interest. To solve for the time variable $t$ when it is in the exponent, we must use logarithms to "bring down" the exponent and isolate the variable.

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Solution

Part (a)

We need to find the time $t$ when the total amount $A(t)$ reaches Rs. 450,000. We start by substituting the target amount into the given function:

$450,000=50,000(1.06{)}^t$

First, we simplify the equation by dividing both sides by $50,000$:

$\frac{450,000}{50,000}=(1.06{)}^t$ $9=(1.06{)}^t$

Next, we take the logarithm of both sides to solve for the exponent $t$:

$\log 9=\log (1.06{)}^t$

Using the power rule of logarithms ($\log x^n=n\log x$), we can move $t$ to the front:

$\log 9=t\log 1.06$

Now, isolate $t$ by dividing by $\log 1.06$:

$t=\frac{\log 9}{\log 1.06}$

Using a calculator to find the final value:

$t\approx 37.71\text{ years}$

Part (b)

The "doubling time" is the time it takes for the initial investment of Rs. 50,000 to double to Rs. 100,000. We set $A(t)=100,000$:

$100,000=50,000(1.06{)}^t$

Divide both sides by $50,000$:

$2=(1.06{)}^t$

Take the logarithm of both sides:

$\log 2=\log (1.06{)}^t$ $\log 2=t\log 1.06$

Isolate $t$:

$t=\frac{\log 2}{\log 1.06}$

Calculating the value:

$t\approx 11.90\text{ years}$

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Key Formulas or Methods Used

• Exponential Growth Function: $A(t)=P(1+r{)}^t$
• Logarithm Power Rule: $\log (a^b)=b\log a$
• Algebraic Isolation: Dividing constants to simplify the exponential base before applying logarithms.

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Summary of Steps

1. Substitute the given future value ($A$) into the growth function.
2. Divide both sides by the initial principal ($50,000$) to isolate the base and exponent.
3. Apply the common logarithm ($\log$) to both sides of the equation.
4. Use the power rule of logarithms to move the variable $t$ out of the exponent.
5. Divide the logarithms to solve for $t$ and round to two decimal places.