An architecture firm are investigating shapes for the roof of a new house design. They model the roof as a prism formed from two sloped rectangles and two triangles. The triangular cross-section of the prism is an isosceles, right-angled triangle of width $w$ and slant-height $s$ as shown below. The prism is open at the bottom.

Due to building regulations, the volume of the roof cavity is required to equal $108{\text{m}}^3$. In order to reduce heat-loss through the roof, the architects plan to make the total surface area of the roof (including all four faces) as small as possible.

1. What is the smallest possible total area for the roof?

2. Why might these dimensions not be an appropriate solution for the house design?

Let the length of the roof be $\ell$. Then the volume of the roof is equal to the area of the triangular cross-section times $\ell$.

Use the above hint to show that

The total area will be $A=s^2+2s\ell$.

Express the area in terms of $s$ - the minimum area will occur when $\frac{dA}{ds}=0$.

Give your answer below