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A rectangular garden of area 832 square feet is to be surrounded on three sides by a brick wall costing $8 per foot and on one side by a fence costing $5 per foot. find the dimensions of the garden such that the cost of the materials is minimized.

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mathmate
Nice problem!

Area of a rectangle = length * width, or length = Area / width
Let the Width be W
then the length is 832/W 
Assume the side W is a fence, and the rest brick.
Total cost, C(W) = $5*W + $8W + 2*$8*(832/W)
Simplifying, C(W)=13W+13312/W

To find the minimum cost, we differentiate the cost with respect to W, and equate the derivative C'(W) to zero.
then C'(W)=13-13312/W^2=0
Rewrite C'(W) as (13W^2-13312)/(W^2)=0, we solve for W and get 
W=sqrt(13312/13)=32
Therefore length, L=832/32=26
Check L*W=26*32=832  ok  [ note L>W, because this costs less $]
Check L=26 and W=32 is the minimum (as opposed to maximum),
we calculate C"(W)=26624/W^3 > 0 which means that W=26 is a minimum for C(W).
So the dimensions of the garden are 26' x 32', with fence on the W=32' dimension.
Just by curiosity, total cost = C(32)=$832, and average cost = $7.17/'
all sound reasonable.







yoodyannapolis

The dimensions of the garden such that the cost of the materials is minimized are 26 feet, 32 feet.

Let the length and the breadth of the rectangular garden be x and y respectively.

Area of a rectangle = [tex]xy[/tex]

832 square feet=[tex]xy[/tex]

⇒[tex]x=\frac{832}{y}[/tex]

Assume the side y is a fence, and the rest brick.

Total cost, [tex]C(y) = $5y + $8y + 2.8\frac{832}{y}[/tex]

⇒[tex]C(y)=13y+\frac{13312}{y}[/tex]

Differentiate the cost with respect to y and equate the derivative C'(y) to zero for maximum or minimum value of y.

[tex]C'(y)=13-\frac{13312}{y^{2} } \\0=13-\frac{13312}{y^{2} } \\[/tex]

[tex]y=\sqrt{\frac{13312}{13} } \\y=32[/tex]

Therefore the length of the rectangular garden is

[tex]x=\frac{832}{32} \\=26[/tex]

And the second derivative is

[tex]C''(y)=\frac{26624}{y^{3} }\\C''(32)=\frac{26624}{32^{3} }\geq 0[/tex]

So the cost is minimum at y=32 feet

Therefore the dimensions of the garden are x=26 feet, y=32feet.

Learn more:https://brainly.com/question/14959254

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