1. State the problem: Gru wants to fence a rectangular dog pen along his house, so no fence is needed on the house side. He needs to find the dimensions that minimize fencing cost with a fixed area determined by wood chip cost. 2. Determine the area: Wood chips cost 1.50 per square foot, total cost is 108. \[ \text{Area} = \frac{108}{1.50} = 72 \text{ square feet} \] 3. Define variables: Let $x$ be the length perpendicular to the house, and $y$ be the length along the house. 4. Express fencing length: Since no fence on the house side, fencing length is \[ P = 2x + y \] 5. Express area constraint: \[ xy = 72 \to y = \frac{72}{x} \] 6. Substitute $y$ in $P$: \[ P = 2x + \frac{72}{x} \] 7. Minimize $P$ by finding derivative and setting to zero: \[ \frac{dP}{dx} = 2 - \frac{72}{x^2} = 0 \to 2 = \frac{72}{x^2} \to x^2 = \frac{72}{2} = 36 \to x = 6 \] 8. Find $y$: \[ y = \frac{72}{6} = 12 \] 9. Verify minimum: Second derivative \[ \frac{d^2P}{dx^2} = \frac{144}{x^3} \text{ which is positive at } x=6 \Rightarrow \text{minimum} \] 10. Final answer: Gru should make the pen 6 feet perpendicular to the house and 12 feet along the house to minimize fencing.