Question

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. What is the largest area that can be enclosed?

Answer 1

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)

$200=3x+4y$

solve for y

$200=3x+4y$

$200-3x=4y$

divide both sides by 4

$y=50-\frac{3x}{4}$

Area of the rectangle = length times width

$Area = 3x \cdot y$

$Area = 3x \cdot (50-\frac{3x}{4})$

$A=150x-\frac{9x^2}{4}$

Now take derivative

$A'=150-\frac{9x}{2}$

Set it =0 and solve for x

$0=150-\frac{9x}{2}$

$150=\frac{9x}{2}$

multiply both sides by 2/9

$x=\frac{100}{3}$

$A''=-\frac{9}{2}$

For any value of x, second derivative is negative

So maximum at x= 100/3

 $A=150x-\frac{9x^2}{4}$ , replace the value of x

$A=150(\frac{100}{3})-\frac{9(\frac{100}{3})^2}{4})$

Area = 2500 square feet is the largest area enclosed