Optimalization Problem having trouble with

[lamaclass]

I have a quiz on these today and wanted to understand how to do them.

1) A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $30/ft. and on the other three sides by a metal fence costing $10/ft. If the area of the garden is 1,000 square ft, find the dimensions of the garden that minimizes the cost.

Would my equation for it be A=10x3*30y with 1000 as the constraint so A'=30x2*30y dy/dx? :?

 

[soroban]

Hello, lamaclass!

Don't know how or where you got those equations . . .

1) A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $30/ft
and on the other three sides by a metal fence costing $10/ft.
If the area of the garden is 1,000 square ft, find the dimensions of the garden that minimizes the cost.

               x
      * = = = = = = = = *
      |                 |
     y|                 |y
      |                 |
      * - - - - - - - - *
               x

\(\displaystyle \text{The back wall is }x\text{ feet of brick at \$30 per foot.}\)
. . \(\displaystyle \text{Its cost is: }\:30x\text{ dollars.}\)

\(\displaystyle \text{The other three sides is }(x + 2y)\text{ feet of metal at \$10 per foot.}\)
. . \(\displaystyle \text{Its cost is: }\:10(x+2y)\text{ dollars.}\)

\(\displaystyle \text{Hence, the total cost is: }\:C \;=\;30x + 10(x+2y) \;=\;40x + 20y\) .[1]

\(\displaystyle \text{The area is 1000: }\;xy \:=\:1000 \quad\Rightarrow\quad y \:=\:\frac{1000}{x}\) .[2]

\(\displaystyle \text{Substitute [2] into [1]: }\;C \;=\;40x + 20\left(\frac{1000}{x}\right) \quad\Rightarrow\quad C \;=\;40x + 20,\!000x^{-1}\)


\(\displaystyle \text{Differentiate and equate to zero: }\;C' \;=\;40 - 20,\!000x^{-2} \:=\:0\)

\(\displaystyle \text{Multiply by }x^2\!:\;\;40x^2 - 20,000 \:=\:0 \quad\Rightarrow\quad x^2 \:=\:500\)


\(\displaystyle \text{Therefore: }\;x \:=\:10\sqrt{5},\;\;y \:=\:20\sqrt{5}\)