1. **Problem statement:** A woman wants to travel from point A to point C on opposite sides of a circular lake with radius $r=3$ km. She can walk along the shore at 8 km/h and row straight across at 4 km/h. We define $x$ as the angle $\angle BAC$ in radians. 2. **Express total time $T(x)$:** - The arc length from A to B along the circle is $r x = 3x$ km. - Walking speed is 8 km/h, so time walking is $\frac{3x}{8}$ hours. - The straight line distance from B to C is the chord length, which is $2r \sin\left(\frac{\pi - x}{2}\right) = 2 \times 3 \times \sin\left(\frac{\pi - x}{2}\right) = 6 \sin\left(\frac{\pi - x}{2}\right)$ km. - Rowing speed is 4 km/h, so time rowing is $\frac{6 \sin\left(\frac{\pi - x}{2}\right)}{4} = \frac{3}{2} \sin\left(\frac{\pi - x}{2}\right)$ hours. Thus, total time: $$ T(x) = \frac{3x}{8} + \frac{3}{2} \sin\left(\frac{\pi - x}{2}\right) $$ 3. **Calculate derivative $T'(x)$:** - Derivative of $\frac{3x}{8}$ is $\frac{3}{8}$. - Derivative of $\frac{3}{2} \sin\left(\frac{\pi - x}{2}\right)$ is $\frac{3}{2} \cos\left(\frac{\pi - x}{2}\right) \times \left(-\frac{1}{2}\right) = -\frac{3}{4} \cos\left(\frac{\pi - x}{2}\right)$. So, $$ T'(x) = \frac{3}{8} - \frac{3}{4} \cos\left(\frac{\pi - x}{2}\right) $$ 4. **Find shortest time:** - Set $T'(x) = 0$: $$ \frac{3}{8} - \frac{3}{4} \cos\left(\frac{\pi - x}{2}\right) = 0 \implies \cos\left(\frac{\pi - x}{2}\right) = \frac{1}{2} $$ - Solve for $x$: $$ \frac{\pi - x}{2} = \frac{\pi}{3} \implies \pi - x = \frac{2\pi}{3} \implies x = \pi - \frac{2\pi}{3} = \frac{\pi}{3} $$ - Calculate $T\left(\frac{\pi}{3}\right)$: $$ T\left(\frac{\pi}{3}\right) = \frac{3 \times \frac{\pi}{3}}{8} + \frac{3}{2} \sin\left(\frac{\pi - \frac{\pi}{3}}{2}\right) = \frac{\pi}{8} + \frac{3}{2} \sin\left(\frac{2\pi/3}{2}\right) = \frac{\pi}{8} + \frac{3}{2} \sin\left(\frac{\pi}{3}\right) $$ - Since $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$, $$ T\left(\frac{\pi}{3}\right) = \frac{\pi}{8} + \frac{3}{2} \times \frac{\sqrt{3}}{2} = \frac{\pi}{8} + \frac{3\sqrt{3}}{4} \approx 0.393 + 1.299 = 1.692 \text{ hours} $$ 5. **Find longest time:** - The longest time occurs when she walks the entire half-circle (no rowing), so $x = \pi$. - Then, $$ T(\pi) = \frac{3 \pi}{8} + \frac{3}{2} \sin\left(\frac{\pi - \pi}{2}\right) = \frac{3 \pi}{8} + 0 = \frac{3 \pi}{8} \approx 1.178 \times 3 = 3.534 \text{ hours} $$ **Final answers:** - (a) $T(x) = \frac{3x}{8} + \frac{3}{2} \sin\left(\frac{\pi - x}{2}\right)$ - (b) $T'(x) = \frac{3}{8} - \frac{3}{4} \cos\left(\frac{\pi - x}{2}\right)$ - (c) Shortest time $\approx 1.692$ hours - (d) Longest time $\approx 3.534$ hours