Given:
The amount for buy fence = $ 10,000.
The cost for the top and bottom of the fence = $ 4 per foot.
The cost for the left ad right sides of the fence = $3 per foot.
The length of the top and bottom is x.
The length of the left and right is y.
Required:
We need to find the money that was used for the top and bottom, x, and for the sides, y.
Explanation:
Consider the equation for the given model.
[tex]4(2x)+3(2y)=10000[/tex][tex]8x+6y=10000[/tex]Isolate y, we get
[tex]8x+6y-8x=10000-8x[/tex][tex]6y=10000-8x[/tex][tex]\frac{6y}{6}=\frac{10000}{6}-\frac{8x}{6}[/tex][tex]y=\frac{5000}{3}-\frac{4x}{3}[/tex]Multiply x and y, since the area of the given rectangle, is xy square feet.
[tex]xy=\frac{5000}{3}x-\frac{4x^2}{3}[/tex]Let A =xy.
[tex]A=\frac{5000}{3}x-\frac{4x^2}{3}[/tex]Differentiate the equation with respect to x.
[tex]A^{\prime}=\frac{5000}{3}-\frac{4(2x)}{3}[/tex][tex]A^{\prime}=\frac{5000}{3}-\frac{8x}{3}[/tex]Equate this equation to zero.
[tex]A^{\prime}=\frac{5000}{3}-\frac{8x}{3}=0[/tex][tex]5000-8x=0[/tex][tex]5000-8x+8x=0+8x[/tex][tex]5000=8x[/tex][tex]\frac{5000}{8}=\frac{8x}{8}[/tex][tex]x=625\text{ feet.}[/tex]Substitute x =625 in the equation y.
[tex]y=\frac{5000}{3}-\frac{4(625)}{3}[/tex][tex]y=\frac{5000}{3}-\frac{2500}{3}[/tex][tex]y=\frac{5000-2500}{3}[/tex][tex]y=\frac{2500}{3}[/tex][tex]y=833.33[/tex][tex]y=833.33feet[/tex]Replace x =625 in 8x to find the cost for the top and bottom.
[tex]8\times625=5000[/tex]Replace y =833.33 in 6y to find the cost for the left and right sides.
[tex]6\times833.33=4999.98[/tex][tex]=5000[/tex][tex]A=xy=625\times833.33=520831.25feet^2[/tex]Final answer:
The area will be x times y or 520931.250square feet.
Top and bottom fencing will cost $5000, and sides will cost $5000.