Question

The money invested in a company is compounded continuously. Rs. 400 invested today becomes Rs. 800 in 6 years, then at the end of 33 years, it will become .. ($\sqrt{2} = 1.4142$)

• A. $9050.88$
• B. $18101.76$
• C. $6788.16$
• D. $12067.84$

Hint:

If money doubles in $T$ years, it becomes $A_0 \cdot 2^{t/T}$ after $t$ years in continuous compounding.

Answer 1

The Correct Option is B

Step 1: Concept
Continuous compounding is modeled by the differential equation $\frac{dA}{dt} = rA$, which leads to the formula $A(t) = A_0 e^{rt}$.

Step 2: Meaning
$A_0$ is the initial investment, $r$ is the rate of interest, and $t$ is time in years.

Step 3: Analysis
Given $A_0 = 400$. At $t = 6$, $A = 800$. $800 = 400 e^{6r} \implies e^{6r} = 2 \implies e^r = 2^{1/6}$. We need $A$ at $t = 33$: $A(33) = 400 (e^{r})^{33} = 400 (2^{1/6})^{33} = 400 (2^{33/6}) = 400 (2^{5.5})$. $A(33) = 400 \times 2^5 \times 2^{0.5} = 400 \times 32 \times \sqrt{2}$. $A(33) = 12800 \times 1.4142$.

Step 4: Conclusion
$12800 \times 1.4142 = 18101.76$. Final Answer: (B)