Step 1: Understanding the Concept:
When money is compounded continuously, the amount $A$ after time $t$ is given by the formula $A = P e^{rt}$, where $P$ is the principal amount and $r$ is the continuous compounding rate.
Step 2: Key Formula or Approach:
Use the formula $A = P e^{rt}$.
First, use the given condition ($P=400$, $A=800$, $t=6$) to find the value of $e^r$.
Then, use this value to calculate the final amount for $t=33$.
Step 3: Detailed Explanation:
Given principal $P = 400$.
After $t = 6$ years, the amount $A = 800$.
Substituting these values into the formula:
\[ 800 = 400 e^{6r} \]
\[ e^{6r} = \frac{800}{400} = 2 \]
To find $e^r$, we take the 6th root of both sides:
\[ (e^{6r})^{1/6} = 2^{1/6} \Rightarrow e^r = 2^{1/6} \]
Now, we need to find the amount at the end of $t = 33$ years.
\[ A_{33} = 400 e^{33r} \]
\[ A_{33} = 400 (e^r)^{33} \]
Substitute $e^r = 2^{1/6}$:
\[ A_{33} = 400 (2^{1/6})^{33} \]
\[ A_{33} = 400 \cdot 2^{33/6} \]
Simplify the fraction $33/6$ by dividing numerator and denominator by 3:
\[ A_{33} = 400 \cdot 2^{11/2} \]
We can write $2^{11/2}$ as $2^5 \cdot 2^{1/2}$:
\[ A_{33} = 400 \cdot 2^5 \cdot \sqrt{2} \]
\[ A_{33} = 400 \cdot 32 \cdot \sqrt{2} \]
\[ A_{33} = 12800 \cdot \sqrt{2} \]
Given $\sqrt{2} = 1.4142$, substitute this value:
\[ A_{33} = 12800 \cdot 1.4142 \]
\[ A_{33} = 128 \cdot 141.42 \]
\[ A_{33} = 18101.76 \]
Step 4: Final Answer:
The amount will become Rs. $18101.76$.