Question: QUESTION 7 h x x A solid cuboid has a square base with length $x$ units and height of $h$ units as shown. The volume of the cuboid is 125 units^3. a) Show that the height of this cuboid is given by $h = \frac{125}{x^2}$ (2 marks) b) Show that the surface area is given by $A(x) = 2x^2 + \frac{500}{x}$ (4 marks) c) Find the value of $x$ such that the cuboid will have minimum surface area. (6 marks) Graph/shape description: A cuboid with a square base is shown in the center; the base edges are labeled $x$ on the bottom front and bottom right, and the vertical height is labeled $h$ on the right side. position_hint: center

1. **Problem Statement:** We have a solid cuboid with a square base of side length $x$ units and height $h$ units. The volume is given as 125 units$^3$. 2. **Part (a): Show that $h = \frac{125}{x^2}$** - Volume of cuboid formula: $$V = \text{base area} \times \text{height} = x^2 \times h$$ - Given volume: $$V = 125$$ - Substitute: $$x^2 h = 125$$ - Solve for $h$: $$h = \frac{125}{x^2}$$ 3. **Part (b): Show that surface area $A(x) = 2x^2 + \frac{500}{x}$** - Surface area of cuboid with square base: $$A = 2(\text{base area}) + \text{perimeter of base} \times \text{height}$$ - Base area = $x^2$ - Perimeter of base = $4x$ - Height = $h = \frac{125}{x^2}$ (from part a) - Substitute: $$A = 2x^2 + 4x \times \frac{125}{x^2} = 2x^2 + \frac{500x}{x^2} = 2x^2 + \frac{500}{x}$$ 4. **Part (c): Find $x$ for minimum surface area** - We want to minimize: $$A(x) = 2x^2 + \frac{500}{x}$$ - Differentiate $A(x)$ with respect to $x$: $$A'(x) = 4x - \frac{500}{x^2}$$ - Set derivative to zero for critical points: $$4x - \frac{500}{x^2} = 0$$ - Multiply both sides by $x^2$: $$4x^3 - 500 = 0$$ - Solve for $x^3$: $$4x^3 = 500$$ $$x^3 = \frac{500}{4} = 125$$ - Take cube root: $$x = \sqrt[3]{125} = 5$$ 5. **Verify minimum by second derivative test:** - Second derivative: $$A''(x) = 4 + \frac{1000}{x^3}$$ - At $x=5$: $$A''(5) = 4 + \frac{1000}{125} = 4 + 8 = 12 > 0$$ - Since $A''(5) > 0$, $x=5$ is a minimum. 6. **Final answer:** - The cuboid has minimum surface area when $x = 5$ units.