Question 7: Abdullah invested Rs. \(100,000\) in a saving scheme and gains interest at the rate of \(5\%\) per annum so that the total value of this investment after \(t\) years is Rs. \(y\). This is modelled by an equation \(y=100,000\ (1.05)^t\), \(t\ge0\). Find after how many years the investment will be double.
Solution:
It is given that
\[y=100,000\ (1.05)^t\]
According to the given condition of the question investment will double. Set \(y=200,000\) in above equation, we have
\[
\begin{align*}
200,000&=100,000\ (1.05)^t\\
\frac{200,000}{100,000}&=(1.05)^t\\
2&=(1.05)^t\\
\end{align*}
\]
Taking logarithm on both sides, we have
\[
\begin{align*}
\log2&=\log(1.05)^t\\
0.3010&=t(0.0212)\\
t&=\frac{0.3010}{0.0212}\\
t&=14.20
\end{align*}
\]
The investemtn will double in approximately 14.20 years.