More Optimization — Question 6

Question 6 - Solution

Let: - $x$ = length along the river (no fence needed) - $y$ = depth away from river (fence on both sides)

Area constraint: $$xy = 1000 \Rightarrow y = \frac{1000}{x}$$

Cost function: - Two sides of length $y$ at $10/m: cost = $2(10)(y) = 20y$ - One side of length $x$ at $5/m: cost = $5x$

$$C(x) = 5x + 20 \cdot \frac{1000}{x} = 5x + \frac{20000}{x}$$

Minimize cost — take derivative: $$C'(x) = 5 - \frac{20000}{x^2}$$

Set derivative to zero: $$5 = \frac{20000}{x^2} \Rightarrow x^2 = 4000 \Rightarrow x = \sqrt{4000} = 20\sqrt{10}$$

Find $y$: $$y = \frac{1000}{x} = \frac{1000}{20\sqrt{10}} = \frac{50}{\sqrt{10}} = 5\sqrt{10}$$

Answer: $$\boxed{ \begin{aligned} x &= 20\sqrt{10} \approx 63.25\ \text{m} \\ y &= 5\sqrt{10} \approx 15.81\ \text{m} \end{aligned} }$$

These dimensions minimize the fencing cost.