Geometric Optimization Math Example 4

Solution

1. 1
   Step 1: Let base side $= x$ and height $= h$. Volume: $x^2 h = 32 \Rightarrow h = 32/x^2$.
2. 2
   Step 2: Surface area: $S = x^2 + 4xh = x^2 + 4x \cdot \dfrac{32}{x^2} = x^2 + \dfrac{128}{x}$.
3. 3
   Step 3: Minimise: $\dfrac{dS}{dx} = 2x - \dfrac{128}{x^2} = 0 \Rightarrow x^3 = 64 \Rightarrow x = 4$ cm.
4. 4
   Step 4: $h = 32/16 = 2$ cm. $S = 16 + 32 = 48$ cm$^2$.