1. **State the problem:** A boatbuilding firm makes two types of dinghy: fiberglass and wooden-hulled. Fiberglass dinghies sell at a profit of 13 each and require 2 craftsmen and 3 apprentices to build. Wooden-hulled dinghies sell at a profit of 20 each and require 3 craftsmen and 5 apprentices. The firm employs 12 craftsmen and 19 apprentices. We need to find the ratio in which the two types of boats should be built to maximize profit. 2. **Define variables:** Let $x$ = number of fiberglass dinghies built Let $y$ = number of wooden-hulled dinghies built 3. **Write constraints based on craftsmen and apprentices:** Craftsmen constraint: $$2x + 3y \leq 12$$ Apprentices constraint: $$3x + 5y \leq 19$$ 4. **Profit function to maximize:** $$P = 13x + 20y$$ 5. **Find feasible points by solving constraints:** From craftsmen constraint: $$2x + 3y \leq 12$$ From apprentices constraint: $$3x + 5y \leq 19$$ 6. **Find intercepts for each constraint:** For craftsmen: - If $x=0$, then $3y=12 \Rightarrow y=4$ - If $y=0$, then $2x=12 \Rightarrow x=6$ For apprentices: - If $x=0$, then $5y=19 \Rightarrow y=\frac{19}{5} = 3.8$ - If $y=0$, then $3x=19 \Rightarrow x=\frac{19}{3} \approx 6.33$ 7. **Find intersection point of the two constraints:** Solve system: $$\begin{cases} 2x + 3y = 12 \\ 3x + 5y = 19 \end{cases}$$ Multiply first equation by 3 and second by 2: $$\begin{cases} 6x + 9y = 36 \\ 6x + 10y = 38 \end{cases}$$ Subtract first from second: $$6x + 10y - (6x + 9y) = 38 - 36 \Rightarrow y = 2$$ Substitute $y=2$ into first equation: $$2x + 3(2) = 12 \Rightarrow 2x + 6 = 12 \Rightarrow 2x = 6 \Rightarrow x = 3$$ 8. **Evaluate profit at corner points:** - At $(0,0)$: $P=0$ - At $(6,0)$: $P=13(6)+20(0)=78$ - At $(0,3.8)$: $P=13(0)+20(3.8)=76$ - At $(3,2)$: $P=13(3)+20(2)=39+40=79$ 9. **Conclusion:** Maximum profit is 79 at $x=3$ fiberglass and $y=2$ wooden-hulled dinghies. 10. **Ratio of fiberglass to wooden-hulled boats:** $$\frac{x}{y} = \frac{3}{2}$$ **Final answer:** The boats should be built in the ratio 3:2 (fiberglass to wooden-hulled) to maximize profit.