1. **Problem statement:** A man is 12 miles south of a straight beach and wants to reach a point 20 miles east along the shore. He can travel by boat at 20 mph by paying 2 dollars per hour for boat usage, and by land at 0.06 dollars per mile. What is the minimum cost to reach his destination? 2. **Define variables:** Let $x$ be the distance (miles) along the beach from the nearest point to the man's starting position to the point where he lands by boat. The boat path is from the starting point to that shore point, and then he travels on land the remaining $20-x$ miles east. 3. **Calculate boat distance:** Boat distance is the hypotenuse of a right triangle with legs 12 (south) and $x$ (along shore). So, boat distance $$d_b = \sqrt{12^2 + x^2} = \sqrt{144 + x^2}$$ miles. 4. **Time and cost for boat:** Speed is 20 mph, time $$t_b = \frac{d_b}{20} = \frac{\sqrt{144 + x^2}}{20}$$ hours. Cost of boat use is $2$ dollars per hour, so cost $$C_b = 2 \times t_b = 2 \times \frac{\sqrt{144 + x^2}}{20} = \frac{\sqrt{144 + x^2}}{10}$$ dollars. 5. **Distance and cost on land:** Land distance is $20 - x$ miles, cost per mile is $0.06$, so $$C_l = 0.06 (20 - x)$$ dollars. 6. **Total cost function:** $$C(x) = C_b + C_l = \frac{\sqrt{144 + x^2}}{10} + 0.06 (20 - x)$$ dollars, for $$0 \leq x \leq 20$$. 7. **Minimize total cost:** Find $x$ minimizing $$C(x)$$ by taking derivative and setting it to zero. \[ C'(x) = \frac{1}{10} \times \frac{x}{\sqrt{144 + x^2}} - 0.06 = 0 \] Solving for $x$: \[ \frac{x}{10 \sqrt{144 + x^2}} = 0.06 \] \[ \frac{x}{\sqrt{144 + x^2}} = 0.6 \] Square both sides: \[ \frac{x^2}{144 + x^2} = 0.36 \] \[ x^2 = 0.36 (144 + x^2) = 51.84 + 0.36 x^2 \] \[ x^2 - 0.36 x^2 = 51.84 \] \[ 0.64 x^2 = 51.84 \] \[ x^2 = \frac{51.84}{0.64} = 81 \] \[ x = 9 \quad \text{(since } x \geq 0) \] 8. **Calculate minimum cost:** Boat distance: $$ d_b = \sqrt{144 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 $$ miles. Boat cost: $$ C_b = \frac{15}{10} = 1.5 $$ dollars. Land cost: $$ C_l = 0.06 (20 - 9) = 0.06 \times 11 = 0.66 $$ dollars. Total cost: $$ C = 1.5 + 0.66 = 2.16 $$ dollars. **Final answer: The man must pay 2.16 dollars for the trip.**