Optimization

[rousie313]

The volume of a building in the shape of a triangle with an equilateral cross-section must be 225,000.
V=1/2(l)(w)(h)
Foundation costs $30 per sq ft
Sides cost $42 per sq ft
Roofing costs $42 per sq ft
Minimize Cost
Need help finding the cost formula to minimize?? stuck here, would appreciate being put in right direction

 

[galactus]

Do you have or did you draw a picture. It would help.

An equilateral triangle has all sides the same length, Hence, the name equialteral.

The height of said equilateral triangle is \(\displaystyle H=\frac{\sqrt{3}}{2}W\)

Therefore, the volume of the building is \(\displaystyle \frac{1}{2}LW(\frac{\sqrt{3}}{2}W)=225,000\).......[1]

The area of an equilateral triangle is \(\displaystyle A=\frac{\sqrt{3}}{4}W^{2}\). Where W is the side length.

The surface area cost will be \(\displaystyle S=\underbrace{30LW}_{\text{base}}+\underbrace{42\cdot 2\cdot \frac{\sqrt{3}}{4}W^{2}}_{\text{ends}}+\underbrace{42\cdot 2\cdot WL}_{\text{roof}}\).

S is what must be minimized. Solve [1] for L or W and sub into [2] so it is reduced to 1 variable. Then, differentiate, set to 0 and solve.

Got it OK now?.