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}](https://tex.z-dn.net/?f=%5CRightarrow%20y%20%3D%20%5Cfrac%7B208%7D%7Bx%7D)
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)]](https://tex.z-dn.net/?f=C%3D%20%20%5B%20%282x%5Ctimes%208%29%2B%28y%5Ctimes%208%29%2B%28y%5Ctimes%205%29%5D)
= (16x+ 8y +5y)
![=16x+13y](https://tex.z-dn.net/?f=%3D16x%2B13y)
![=16x+\frac{13\times 208}{x}](https://tex.z-dn.net/?f=%3D16x%2B%5Cfrac%7B13%5Ctimes%20208%7D%7Bx%7D)
![=16x +\frac{2704}{x}](https://tex.z-dn.net/?f=%3D16x%20%2B%5Cfrac%7B2704%7D%7Bx%7D)
To find the maximum or minimum point, we need to find out
and set
=0.
![\frac{dC}{dx}=16 -\frac{2704}{x^2}](https://tex.z-dn.net/?f=%5Cfrac%7BdC%7D%7Bdx%7D%3D16%20-%5Cfrac%7B2704%7D%7Bx%5E2%7D)
Then
![16 -\frac{2704}{x^2}=0](https://tex.z-dn.net/?f=16%20-%5Cfrac%7B2704%7D%7Bx%5E2%7D%3D0)
![\Rightarrow \frac{2704}{x^2}=16](https://tex.z-dn.net/?f=%5CRightarrow%20%5Cfrac%7B2704%7D%7Bx%5E2%7D%3D16)
![\Rightarrow x^2 =\frac{2704}{16}](https://tex.z-dn.net/?f=%5CRightarrow%20x%5E2%20%3D%5Cfrac%7B2704%7D%7B16%7D)
![\Rightarrow x=\pm \sqrt{169}](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D%5Cpm%20%5Csqrt%7B169%7D)
![\Rightarrow x=\pm 13](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D%5Cpm%2013)
Again ![\frac{d^2C}{dx}= \frac {8112}{x^3}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2C%7D%7Bdx%7D%3D%20%20%5Cfrac%20%7B8112%7D%7Bx%5E3%7D)
![\therefore\left| \frac{d^2C}{dx}\right |_{x=13}= \frac {8112}{13^3}>0](https://tex.z-dn.net/?f=%5Ctherefore%5Cleft%7C%20%5Cfrac%7Bd%5E2C%7D%7Bdx%7D%5Cright%20%7C_%7Bx%3D13%7D%3D%20%20%5Cfrac%20%7B8112%7D%7B13%5E3%7D%3E0)
Therefore at x= 13 , the cost is minimum.
Therefore x = 13 feet.
The other side of the garden is
feet = 16 feet.
Therefore the dimensions of the garden is 16 feet by 13 feet.