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Sagot :
Step-by-step explanation:
Let the side parallel to the river be x meter and the other two sides be of y meter, then
[tex]12 x+(2 y)(4)=3600[/tex]
[tex]12 x+8 y &=3600[/tex]
[tex]3 x+2 y &=900[/tex]
[tex]2 y &=900-3 x[/tex]
The area of the rectangle is [tex]A=xy [/tex]
Substitute into to express the area in a single variable x as,
[tex]A(x) &=x\left[\frac{1}{2}(900-3 x)\right][/tex]
[tex]=\frac{1}{2}\left(900 x-3 x^{2}\right)[/tex]
Differentiate A(x) with respect to x and equate to zero as,
[tex]A^{\prime}(x) &=0[/tex]
[tex]\frac{d}{d x}\left[\frac{1}{2}\left(900 x-3 x^{2}\right)\right]=0[/tex]
[tex]900-3(2 x) =0[/tex]
x=150
Here, [tex]A^{\prime \prime}(x)=-3 x<0[/tex] at [tex]x=150 \Rightarrow A[/tex] is maximum
Therefore, the dimension is:
Length of side parallel to the river: x=150 m.
Length of other two side: [tex]y=\frac{1}{2}(900-450)=225[/tex] m.
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