Answer :
You have d dollars to buy fence to enclose a rectangular plot of land. the fence for the top and bottom costs $a per foot and for the sides it costs $b per foot.
To find the dimensions of the plot with the largest area.
Total Money = D dollars
Total cost for top and bottom fence = $4(2x)
Total cost for sides fence = $8(2y)
Hence total cost of fencing = 4(2x)+8(2y) = 8x+16y
C = 8x+16y
Given cost (c) = D dollars
D = 8x+16y
Now Area of the plot A = xy
A = xy is optimizing function and D = 8x+16y
= [tex](\frac{D-16y}{8})=x[/tex]
A = [tex]\frac{D-16y}{8}y[/tex]
For A to be maximum [tex]\frac{dA}{dy}=0[/tex]
A = [tex]\frac{D}{8}y-\frac{16}{8}y^{2}[/tex]
[tex]\frac{dA}{dy} = \frac{D}{8}-\frac{16}{8}(2y)[/tex]
[tex]\frac{dA}{dy} = \frac{D}{8}-4y[/tex]
[tex]\frac{dA}{dy} = 0 = \frac{D}{8}-4y=0[/tex]
[tex]\frac{D}{8} = 4y = y = \frac{D}{32}[/tex]
Now [tex]\frac{d^{2}A }{dy^{2}} = -4 = -ve number[/tex]
for y = [tex]\frac{D}{32}[/tex] area is maximum
[tex]x = \frac{D-16y}{8} = \frac{D-16 (\frac{D}{32} )}{8} = \frac{D}{16}[/tex]
Say the top and bottom dimensions are = [tex]\frac{D}{16}[/tex]
Sides dimensions are = [tex]\frac{D}{32}[/tex]
To learn more about dimensions click here https://brainly.com/question/28464
#SPJ4
