1. **Problem Statement:** Calculate the hours of time taken by the workman to complete the work order given the Rowan Plan bonus system. 2. **Given Data:** - Time allotted for the job, $T = 40$ hours - Normal rate of wages, $R = 125$ per hour - Factory overhead charges, $F = 50$ per hour - Factory cost of work order, $C = 1700$ - Cost of material, $M = 1000$ 3. **Formula and Explanation:** Under the Rowan Plan, the bonus is calculated based on the ratio of time taken to time allotted. The total factory cost is the sum of wages, overhead, and material cost: $$C = \text{Wages} + \text{Overhead} + M$$ Wages paid to the worker when time taken is $t$ hours: $$\text{Wages} = R \times t \times \left(1 - \frac{t}{2T}\right)$$ Overhead cost: $$\text{Overhead} = F \times t$$ 4. **Set up the equation:** $$C = R t \left(1 - \frac{t}{2T}\right) + F t + M$$ Substitute the known values: $$1700 = 125 t \left(1 - \frac{t}{80}\right) + 50 t + 1000$$ 5. **Simplify the equation:** $$1700 = 125 t - \frac{125 t^2}{80} + 50 t + 1000$$ Combine like terms: $$1700 = 175 t - \frac{125 t^2}{80} + 1000$$ 6. **Bring all terms to one side:** $$175 t - \frac{125 t^2}{80} + 1000 - 1700 = 0$$ $$175 t - \frac{125 t^2}{80} - 700 = 0$$ Multiply through by 80 to clear denominator: $$80 \times 175 t - 125 t^2 - 80 \times 700 = 0$$ $$14000 t - 125 t^2 - 56000 = 0$$ Rearranged: $$-125 t^2 + 14000 t - 56000 = 0$$ Multiply entire equation by -1: $$125 t^2 - 14000 t + 56000 = 0$$ 7. **Solve quadratic equation:** $$125 t^2 - 14000 t + 56000 = 0$$ Divide entire equation by 5 for simplicity: $$25 t^2 - 2800 t + 11200 = 0$$ Calculate discriminant $\Delta$: $$\Delta = (-2800)^2 - 4 \times 25 \times 11200 = 7840000 - 1120000 = 6720000$$ Calculate roots: $$t = \frac{2800 \pm \sqrt{6720000}}{2 \times 25}$$ $$\sqrt{6720000} \approx 2591.19$$ So, $$t_1 = \frac{2800 + 2591.19}{50} = \frac{5391.19}{50} = 107.82$$ $$t_2 = \frac{2800 - 2591.19}{50} = \frac{208.81}{50} = 4.18$$ 8. **Interpretation:** Since the time allotted is 40 hours, the time taken $t$ must be less than or equal to 40. Therefore, the valid solution is: $$\boxed{t = 4.18 \text{ hours}}$$ This means the workman took approximately 4.18 hours to complete the work order under the Rowan Plan.