1. **State the problem:** A man is 12 miles south of a straight beach and needs to reach a point 20 miles east along the shore. He can travel by motorboat to some point on the beach and then walk along the beach to his destination. The boat speed is 20 mph costing 2 dollars per hour, and walking cost is 0.06 dollars per mile. We want to find the minimum total cost. 2. **Set up variables:** Let $x$ be the distance east from the point directly north on the shore where he lands the boat. So, the boat travels from the man’s position to the landing point, and then he walks the remaining distance $20 - x$ along the shore. 3. **Express travel distances:** The boat distance is the straight-line distance from his position (0, -12) to the landing point $(x,0)$: $$d_{boat} = \sqrt{x^2 + 12^2} = \sqrt{x^2 + 144}$$ The walking distance along the beach is $$d_{walk} = 20 - x$$ 4. **Calculate travel times:** Boat time: $$t_{boat} = \frac{d_{boat}}{20} = \frac{\sqrt{x^2 + 144}}{20}$$ Walking cost is per mile, so no time calculation is needed for walking. 5. **Calculate costs:** - Boat cost: $$C_{boat} = 2 \times t_{boat} = 2 \times \frac{\sqrt{x^2 + 144}}{20} = \frac{\sqrt{x^2 + 144}}{10}$$ - Walking cost: $$C_{walk} = 0.06 \times d_{walk} = 0.06(20 - x)$$ 6. **Total cost function:** $$C(x) = \frac{\sqrt{x^2 + 144}}{10} + 0.06 (20 - x)$$ 7. **Minimize total cost:** Take derivative and set to zero: $$C'(x) = \frac{1}{10} \times \frac{x}{\sqrt{x^2 + 144}} - 0.06 = 0$$ $$\Rightarrow \frac{x}{10 \sqrt{x^2 + 144}} = 0.06$$ Multiply both sides by $10 \sqrt{x^2 + 144}$: $$x = 0.6 \sqrt{x^2 + 144}$$ Square both sides: $$x^2 = 0.36 (x^2 + 144)$$ $$x^2 = 0.36 x^2 + 51.84$$ Subtract $0.36 x^2$: $$x^2 - 0.36 x^2 = 51.84$$ $$0.64 x^2 = 51.84$$ $$x^2 = \frac{51.84}{0.64} = 81$$ $$x = 9$$ (take positive because distance). 8. **Calculate total cost at $x=9$:** Boat distance: $$\sqrt{9^2 + 144} = \sqrt{81 + 144} = \sqrt{225} = 15$$ Boat cost: $$\frac{15}{10} = 1.5$$ Walking cost: $$0.06 \times (20 - 9) = 0.06 \times 11 = 0.66$$ Total cost: $$1.5 + 0.66 = 2.16$$ **Answer:** The man must pay $2.16 for the trip.