1. **Problem Statement:** A man is on an island 1 km from the shore. The shore is a straight horizontal line, and the island is 1 km perpendicular above the shore. The ranger's office is 4 km along the shore from the nearest point to the island. The man can paddle at 5 km/h and run at 10 km/h. We want to find the point $x$ km along the shore where he should land his canoe to minimize total travel time to the office. 2. **Set up variables and formula:** - Let $x$ be the distance from the nearest shore point to where he lands the canoe. - Distance paddled is the hypotenuse of a right triangle with legs 1 km (vertical) and $x$ km (horizontal): $$\text{paddle distance} = \sqrt{1^2 + x^2} = \sqrt{1 + x^2}$$ - Distance run is from landing point $x$ to office at 4 km: $$\text{run distance} = 4 - x$$ 3. **Speeds:** - Canoe speed = 5 km/h - Running speed = 10 km/h 4. **Total time function:** $$ T(x) = \frac{\sqrt{1 + x^2}}{5} + \frac{4 - x}{10} $$ 5. **Find minimum time:** - Differentiate $T(x)$ with respect to $x$: $$ T'(x) = \frac{1}{5} \cdot \frac{x}{\sqrt{1 + x^2}} - \frac{1}{10} $$ - Set derivative to zero to find critical points: $$ \frac{x}{5\sqrt{1 + x^2}} = \frac{1}{10} \\ \Rightarrow 2x = \sqrt{1 + x^2} $$ - Square both sides: $$ 4x^2 = 1 + x^2 \\ 3x^2 = 1 \\ x^2 = \frac{1}{3} \\ x = \frac{1}{\sqrt{3}} \approx 0.577 $$ 6. **Check endpoints and critical point:** - At $x=0$, $T(0) = \frac{\sqrt{1}}{5} + \frac{4}{10} = 0.2 + 0.4 = 0.6$ hours - At $x=4$, $T(4) = \frac{\sqrt{1 + 16}}{5} + 0 = \frac{\sqrt{17}}{5} \approx 0.824$ hours - At $x=0.577$, $$ T(0.577) = \frac{\sqrt{1 + 0.577^2}}{5} + \frac{4 - 0.577}{10} = \frac{\sqrt{1.333}}{5} + \frac{3.423}{10} \\ = \frac{1.155}{5} + 0.3423 = 0.231 + 0.3423 = 0.5733 \text{ hours} $$ 7. **Conclusion:** The minimum time is approximately 0.573 hours when the man lands his canoe about 0.577 km down the shore from the nearest point. --- **Diagram description:** - Horizontal shore line with points marked at 0 (nearest shore point) and 4 (ranger's office). - Vertical line from shore at 0 up to island at 1 km. - Canoe path from island to point $x$ on shore. - Running path from $x$ to 4 km.