Question

If \(\int \frac{sinx}{sin4x}\,dx = αlog\frac{1+sinx}{1-sinx}+βlog\frac{1+\sqrt{2}sinx}{1-\sqrt{2}sinx}+c\), then the value of \(32(α+β^2) =\)

• A. \(5\)
• B. \(-1\)
• C. \(9\)
• D. \(-3\)

Hint:

If volume loss is proportional to surface area, the radius decreases at a constant rate.

Answer 1

The Correct Option is D

Step 1: Integrate:
$\frac{dr}{dt} = -k$ gives $r = r_0 - kt = 3 - kt$.

Step 2: Find k and the end time:
At $t = 4$, $r = 1$: $1 = 3 - 4k$, so $k = \frac12$. The ball vanishes when $r = 0$: $0 = 3 - \frac t2$, so $t = 6$ months (A).

Final Answer:
6 months. \[ \boxed{6\text{ months}} \]