1. **State the problem:** Kailash manufactures a cuboid container with a square base of side length $l$ cm and height $d$ cm, where $d$ is three times $l$. The container holds 375 cm³. We need to find $l$, $d$, and the total external surface area. 2. **Expression for $d$ in terms of $l$:** Given height $d$ is three times the base length $l$, so $$d = 3l$$ 3. **Find $l$ and $d$ given volume 375 cm³:** Volume of cuboid $V = l^2 \times d$ Substitute $d = 3l$: $$V = l^2 \times 3l = 3l^3$$ Given $V = 375$, so $$3l^3 = 375$$ Divide both sides by 3: $$\cancel{3}l^3 = \frac{375}{\cancel{3}}$$ $$l^3 = 125$$ Take cube root: $$l = \sqrt[3]{125} = 5$$ Now find $d$: $$d = 3l = 3 \times 5 = 15$$ 4. **Calculate total external surface area of the cuboid:** Surface area $S = 2 \times$ (area of base + area of sides) Base is square: area $= l^2 = 5^2 = 25$ Sides: 4 rectangles each $l \times d = 5 \times 15 = 75$ Total side area $= 4 \times 75 = 300$ Total surface area: $$S = 2(25 + 300) = 2 \times 325 = 650$$ 5. **Expression for height $h$ of cylinder in terms of radius $r$:** Volume of cylinder $V = \pi r^2 h$ Given $V = 375$, solve for $h$: $$h = \frac{375}{\pi r^2}$$ 6. **Show that total surface area $A = 2\pi r^2 + \frac{750}{r}$:** Surface area of cylinder: $$A = 2\pi r^2 + 2\pi r h$$ Substitute $h$: $$A = 2\pi r^2 + 2\pi r \times \frac{375}{\pi r^2} = 2\pi r^2 + \frac{750}{r}$$ 7. **Find derivative $\frac{dA}{dr}$:** $$\frac{dA}{dr} = \frac{d}{dr} \left(2\pi r^2 + \frac{750}{r}\right) = 4\pi r - \frac{750}{r^2}$$ 8. **Find $r$ that minimizes $A$:** Set derivative to zero: $$4\pi r - \frac{750}{r^2} = 0$$ Multiply both sides by $r^2$: $$4\pi r^3 = 750$$ Divide both sides by $4\pi$: $$r^3 = \frac{750}{4\pi} = \frac{375}{2\pi}$$ Take cube root: $$r = \sqrt[3]{\frac{375}{2\pi}}$$ 9. **Find minimum value of $A$:** Substitute $r$ back into $A$: $$A = 2\pi r^2 + \frac{750}{r}$$ Use $r^3 = \frac{375}{2\pi}$ to simplify if needed. 10. **Compare total material needed including additional surface area:** Cuboid total material: $$650 + 10\% = 650 \times 1.1 = 715$$ Cylinder total material: Calculate $A$ at minimum $r$, then add 25%: $$A_{min} \times 1.25$$ Compare values to decide which container requires less material. **Final answers:** - $d = 3l$ - $l = 5$ cm, $d = 15$ cm - Cuboid surface area = 650 cm² - Cylinder height $h = \frac{375}{\pi r^2}$ - Surface area $A = 2\pi r^2 + \frac{750}{r}$ - $\frac{dA}{dr} = 4\pi r - \frac{750}{r^2}$ - Minimizing $r = \sqrt[3]{\frac{375}{2\pi}}$ - Minimum $A$ found by substitution - Choose container with less total material after adding extra percentages