1. **Problem statement:** We have a rectangular metal sheet with length twice its width. Squares of side 60 cm are cut from each corner to form an open rectangular tank. The tank's volume is 1920 litres (which is 1920000 cm³ since 1 litre = 1000 cm³). The width of the sheet is $x$ meters. 2. **Express volume in terms of $x$:** - Width of sheet = $x$ m = $100x$ cm - Length of sheet = $2x$ m = $200x$ cm - After cutting 60 cm squares from each corner, the tank's base dimensions become: - Width = $100x - 2 \times 60 = 100x - 120$ cm - Length = $200x - 2 \times 60 = 200x - 120$ cm - Height of tank = 60 cm Volume $V$ of tank = length $\times$ width $\times$ height $$V = (200x - 120)(100x - 120)(60)$$ 3. **Given volume = 1920000 cm³, set up equation:** $$ (200x - 120)(100x - 120)(60) = 1920000 $$ 4. **Simplify:** Divide both sides by 60: $$ \cancel{(200x - 120)(100x - 120)\cancel{(60)}} = \frac{1920000}{\cancel{60}} $$ $$ (200x - 120)(100x - 120) = 32000 $$ 5. **Expand left side:** $$ (200x - 120)(100x - 120) = 200x \times 100x - 200x \times 120 - 120 \times 100x + 120 \times 120 $$ $$ = 20000x^2 - 24000x - 12000x + 14400 $$ $$ = 20000x^2 - 36000x + 14400 $$ 6. **Form quadratic equation:** $$ 20000x^2 - 36000x + 14400 = 32000 $$ Move 32000 to left: $$ 20000x^2 - 36000x + 14400 - 32000 = 0 $$ $$ 20000x^2 - 36000x - 17600 = 0 $$ 7. **Divide entire equation by 4000 to simplify:** $$ \cancel{20000}x^2 - \cancel{36000}x - \cancel{17600} = 0 $$ $$ 5x^2 - 9x - 4.4 = 0 $$ 8. **Solve quadratic using formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=5$, $b=-9$, $c=-4.4$ Calculate discriminant: $$ \Delta = (-9)^2 - 4 \times 5 \times (-4.4) = 81 + 88 = 169 $$ Calculate roots: $$ x = \frac{9 \pm \sqrt{169}}{10} = \frac{9 \pm 13}{10} $$ Two solutions: - $$ x = \frac{9 + 13}{10} = \frac{22}{10} = 2.2 $$ - $$ x = \frac{9 - 13}{10} = \frac{-4}{10} = -0.4 $$ (discard negative) So, width $x = 2.2$ m. 9. **Find length and width of sheet:** - Width = $2.2$ m - Length = $2 \times 2.2 = 4.4$ m 10. **Calculate cost and selling price:** - Area of sheet = length $\times$ width = $4.4 \times 2.2 = 9.68$ m² - Cost of metal sheet = $9.68 \times 1000 = 9680$ - Labour cost = $300 \times 6 = 1800$ - Total cost = $9680 + 1800 = 11480$ - Desired profit = 30% of total cost = $0.3 \times 11480 = 3444$ - Selling price = total cost + profit = $11480 + 3444 = 14924$ **Final answers:** - (a) Volume in terms of $x$ cm: $$V = 60(200x - 120)(100x - 120)$$ - (b) Width = 2.2 m, Length = 4.4 m - (c) Selling price = 14924