Hello, NAPA55!

The answers come out rather ugly . . .

A builder must completely fence in 3 adjacent rectangular lots along a roadway.

Each lot must have an area of 675 square metres.

Fencing along the roadway costs $25/metre while side and back fencing costs $10/metre.

Find the dimensions of each lot that minimize the total cost of fencing. Code:

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

There are $\displaystyle 3x$ m of fencing along the road.

At $25/m, its cost is: .$\displaystyle (25)(3x) \:= \:75x$ dollars.

There are $\displaystyle 3x$ m of back fencing and $\displaystyle 4y$ m of side fencing.

At $10/m, its cost is: .$\displaystyle 10(3x + 4y) \:=\:30x + 40y$ dollars.

The total cost is: .$\displaystyle C \;=\;75x + (30x + 40y) \;=\;105x + 40y$ dollars. .[1]

Each lot has an area of 675 mē: .$\displaystyle xy \:=\:675\quad\Rightarrow\quad y \:=\:\frac{675}{x}$

Substitute into [1]: .$\displaystyle C \;=\;105x + 40\left(\frac{675}{x}\right) \quad\Rightarrow\quad C \;=\;105x + 25,000x^{-1}$

And __that__ is the function we must minimize . . .