Respuesta :
Answer:
Area = 2500 square feet is the largest area enclosed
Step-by-step explanation:
A rectangular piece of land borders a wall. The land is to be enclosed and to be into divided 3 equal plots with 200 feet of fencing
Let x be the length of each box and y be the width of the box
Perimeter of the box= 3(length ) + 4(width)
[tex]200=3x+4y[/tex]
solve for y
[tex]200=3x+4y[/tex]
[tex]200-3x=4y[/tex]
divide both sides by 4
[tex]y=50-\frac{3x}{4}[/tex]
Area of the rectangle = length times width
[tex]Area = 3x \cdot y[/tex]
[tex]Area = 3x \cdot (50-\frac{3x}{4})[/tex]
[tex]A=150x-\frac{9x^2}{4}[/tex]
Now take derivative
[tex]A'=150-\frac{9x}{2}[/tex]
Set it =0 and solve for x
[tex]0=150-\frac{9x}{2}[/tex]
[tex]150=\frac{9x}{2}[/tex]
multiply both sides by 2/9
[tex]x=\frac{100}{3}[/tex]
[tex]A''=-\frac{9}{2}[/tex]
For any value of x, second derivative is negative
So maximum at x= 100/3
[tex]A=150x-\frac{9x^2}{4}[/tex] , replace the value of x
[tex]A=150(\frac{100}{3})-\frac{9(\frac{100}{3})^2}{4})[/tex]
Area = 2500 square feet is the largest area enclosed
