A landscape architect wishes to enclose a rectangular garden on one side by a brick wall costing $24 per foot and on the other three sides by a metal fence costing $8 per foot. If the area of the garden is 800 ft, find the dimensions of the garden minimizing the cost. (Let x be the length of the brick wall and y be the length of an adjacent side in feet.)

The length of the brick wall is 20 ft and the length of an adjacent side is 40 ft and this can be determined by forming the linear equation in two variables.

Given :

A brick wall costs $24 per foot and on the other three sides by a metal fence costs $8 per foot.
The area of the garden is 800 [tex]\rm ft^2[/tex].

Let 'x' be the length of the brick wall and 'y' be the length of an adjacent side in feet. Then the area of the garden is:

xy = 800 --- (1)

The perimeter of the rectangular garden will be:

Perimeter = 24x + 8x +2(8)y ---- (2)

Now, solve equation (1) for y.

[tex]y = \dfrac{800}{x}[/tex] --- (3)

Now, put the value of 'y' in equation (2).

[tex]\rm P=32x+16\times \dfrac{800}{x}[/tex]

Now, for minimizing cost differentiate the perimeter with respect to 'x'.

[tex]\rm P'=32-16\times \dfrac{800}{x^2}[/tex]

Now, equate the above equation to zero.

[tex]0=32-16\times \dfrac{800}{x^2}[/tex]

[tex]\dfrac{800}{2}={x^2}[/tex]

x = 20

Now, put the value of 'x' in the equation (3).

[tex]y = \dfrac{800}{20}[/tex]

y = 40

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