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ollegr [7]
3 years ago
6

A rectangular garden of area 208 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.
Mathematics
1 answer:
vova2212 [387]3 years ago
4 0

Answer:

Therefore the dimensions of the garden is 16 feet by 13 feet.

Step-by-step explanation:

Let the length of the garden be x and the width of the garden be y.

Given that the area of the rectangular garden is 208 square feet.

Therefore,

xy =208

\Rightarrow y = \frac{208}{x}

Again given that,

The garden is to be surrounded on three sides by a brick wall costing $ 8 per feet and the remaining sides by a fence costing $5 per feet.

The perimeter of the rectangle is = 2(length+ breadth)

                                                       = 2(x+y)

                                                       =2x+2y

The total cost of fence

C=  [ (2x\times 8)+(y\times 8)+(y\times 5)]

   = (16x+ 8y +5y)

   =16x+13y

  =16x+\frac{13\times 208}{x}

 =16x +\frac{2704}{x}

To find the maximum or minimum point, we need to find out \frac{dC}{dx} and set \frac{dC}{dx} =0.

\frac{dC}{dx}=16 -\frac{2704}{x^2}

Then

16 -\frac{2704}{x^2}=0

\Rightarrow \frac{2704}{x^2}=16

\Rightarrow x^2 =\frac{2704}{16}

\Rightarrow x=\pm \sqrt{169}

\Rightarrow x=\pm 13

Again \frac{d^2C}{dx}=  \frac {8112}{x^3}

\therefore\left| \frac{d^2C}{dx}\right |_{x=13}=  \frac {8112}{13^3}>0

Therefore at x= 13 , the cost is minimum.

Therefore x = 13 feet.

The other side of the garden is =\frac{208}{13} feet = 16 feet.

Therefore the dimensions of the garden is 16 feet by 13 feet.

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