Answer is: option2
\(\frac{da}{dt} = -3 \frac{db}{dt}\)Solution:
We are given that the sum of \( a \) and three times \( b \) is constant:
\( a + 3b = C \)
where \( C \) is a constant.
Since \( C \) is constant, we differentiate both sides of the equation:
\( \frac{d}{dt} (a + 3b) = \frac{d}{dt} (C) \)
Since the derivative of a constant is zero, this simplifies to:
\( \frac{da}{dt} + 3 \frac{db}{dt} = 0 \)
From the given answer choices:
- (A) \( \frac{da}{dt} = 3 \frac{db}{dt} \) → Incorrect.
- (B) \( \frac{da}{dt} = -3 \frac{db}{dt} \) → Correct.
- (C) \( 3 \frac{da}{dt} + \frac{db}{dt} = 0 \) → Incorrect. because the coefficients are swapped.
- (D) \( \frac{da}{dt} + 3 \frac{db}{dt} = F \), where \( F \) is a function of \( t \) → Incorrect since \( F = 0 \) in our case.
Thus, the correct answer is:
(B) \( \frac{da}{dt} = -3 \frac{db}{dt} \)