Respuesta :
Answer:
Therefore the dimensions of the rectangular garden are 23.42 ft by 20.5 ft.
Step-by-step explanation:
Given that,
A rectangular garden of area 480 square feet.
Let length of the rectangular garden be x which is surrounded by fence and width of the rectangular garden be y.
Then xy is the area of the given rectangular garden .
Then,
xy= 480
[tex]\Rightarrow y=\frac{480}{x}[/tex]
The length of the tree sides which are surrounded by brick wall is = 2y+x.
The cost for the brick wall is =Length×cost per feet= $12(2y+x)
The cost for the fencing is =Length×cost per feet= $ 9x
[tex]\therefore C=12(2y+x)+9x[/tex]
Now putting [tex]y=\frac{480}{x}[/tex]
[tex]\therefore C=12(2.\frac{480}{x}+x)+9x[/tex]
[tex]\Rightarrow C=\frac{11520}{x}+21x[/tex]
Differentiating with respect to x
[tex]C'=-\frac{11520}{x^2}+21[/tex]
Again differentiating with respect to x
[tex]C''=\frac{23040}{x^3}[/tex]
Now we set C'=0
[tex]\therefore-\frac{11520}{x^2}+21=0[/tex]
[tex]\Rightarrow\frac{11520}{x^2}=21[/tex]
[tex]\Rightarrow x^2=\frac{11520}{21}[/tex]
[tex]\Rightarrow x\approx 23.42[/tex]
[tex]C''|_{x=23.42}=\frac{23040}{(23.42)^3}>0[/tex].
Since at x=23.42,C''>0. So at x=23.42, the total cost will be minimum.
The width of the rectangular garden is [tex]y=\frac{480}{x}[/tex]
[tex]=\frac{480}{23.42}[/tex]
[tex]\approx 20.5[/tex]
Therefore the dimensions of the rectangular garden are 23.42 ft by 20.5 ft.
The cost of the material is [tex]C=\frac{11520}{x}+21x[/tex]
[tex]=\frac{11520}{23.42}+21\times 23.42[/tex]
=$983.70
