An amount A (in Rs.) grows so that \(\displaystyle\frac{dA}{dt} = 0.1A\), where t is the time in years, and A = 500 at t = 0. Which of the following statements are correct?
(A) The differential equation is of order 1 and degree 2.
(B) The amount after 10 years is Rs. 500e.
(C) The amount after 20 years is Rs. 1000.
(D) At t = 0 the amount is increasing at the rate of Rs. 50 per year.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B) and (D) only
By separation of variables:
\( \displaystyle \frac{dA}{A} = 0.1\,dt \)
\( \log A = 0.1t + c \)
\( A = 500e^{0.1t} \) (using A = 500 at t = 0)
\( A(10) = 500e^{1} = 500e \)
\( A(20) = 500e^{2} \approx 3695 \)
At t = 0: \( \displaystyle \frac{dA}{dt} = 0.1 \times 500 = 50 \)
So A = 500e0.1t, A(10) = 500e, A(20) = 500e2, the initial rate is Rs. 50 per year, and the equation has order 1 and degree 1.
Checking the statements
(A) Order 1 and degree 2 — Incorrect. dA/dt appears only to the first power, so the degree is 1.
(B) Amount after 10 years is Rs. 500e — Correct. \(500e^{0.1 \times 10} = 500e\).
(C) Amount after 20 years is Rs. 1000 — Incorrect. It is \(500e^{2}\), about Rs. 3695.
(D) Initial rate Rs. 50 per year — Correct. \(0.1 \times 500 = 50\).