Question

Jane has 200 feet of fencing to enclose a rectangular vegetable garden. One side of the garden will be alongside Jane’s house so only three sides will need to be fenced. What is the largest area that can be enclosed by the 200 feet of fencing?

1- 2500 square feet
2- 5000 square feet
3- 4444 square feet
4- 10,000 square feet

Answer 1

Answer:

4- 10,000 square feet.

Step-by-step explanation:

Let x represent length alongside Jane’s house and y represent width of garden.

We have been given that Jana has 200 feet of fencing to enclose a rectangular vegetable garden. One side of the garden will be alongside Jane’s house so only three sides will need to be fenced.

The fencing on 3 sides of garden will be equal to perimeter of garden on three sides.

$P=x+2y$

$200=x+2y$

We know that area of rectangle is equal to length times width that is $A=xy$.

From perimeter equation, we will get:

$x=200-2y$

Substituting this value in area equation, we will get:

$A(y)=(200-y)y$

Now, we need to find derivative of area of function.

$A(y)=200y-y^2$

$A'(y)=200-2y$

Now, we will set derivative equal to 0 to find critical points.

$200-2y=0$

$200=2y$

$\frac{200}{2}=y$

$y=100$

The area will be maximum, when $y=100$.

$A(y)=200y-y^2$

$A(100)=200(100)-(1000)^2$

$A(100)=20,000-10,000$

$A(100)=10,000$

Therefore, the maximum area would be 10,000 square feet and 4th option is correct choice.