Hello, stepho_145!

Here's the first one . . .

1. **Minimizing Construction Costs**

The UNICO department store has decided to enclose an 800 ft² area

outside the building. One side will be formed by the external wall of the store;

two sides will be made of pine boards, and the fourth side will be galvanized steel fencing.

If the pine boards cost $6/ft and the steel fencing costs $3/running foot,

determine the dimensions of the enclosure that can be erected at minimum cost. Code:

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x

The area will be 800 ft².

We have: .$\displaystyle xy \:=\:800\quad\Rightarrow\quad y \:=\:\frac{800}{x}$ .[1]

The cost will be: .$\displaystyle \begin{array}{ccc}x\text{ ft of steel fencing at \$3/ft} & = &3x\text{ dollars} \\

2y\text{ ft of pine fencing at \$6/ft} &=& 12y\text{ dollars} \end{array}$

Hence, the total cost is: .$\displaystyle C \;=\;3x + 12y$ .[2]

Substitute [1] into [2]: . $\displaystyle C \;=\;3x + 12\left(\frac{800}{x}\right)$

Therefore: .$\displaystyle C \;=\;3x + 9600x^{-1}$

. . and **that** is the function we must minimize.

*Go for it!*