mary needs to enclose a field with a rectangular fence. she has 500 ft of fencing material and a building is…

mary needs to enclose a field with a rectangular fence. she has 500 ft of fencing material and a building is on one side of the field and therefore wont need any fencing. the dimensions of the field that encloses the largest area are ______ feet and ______ feet.

mary needs to enclose a field with a rectangular fence. she has 500 ft of fencing material and a building is on one side of the field and therefore wont need any fencing. the dimensions of the field that encloses the largest area are ______ feet and ______ feet.

Answer

Explanation:

Step1: Define variables

Let the length of the side parallel to the building be $x$ and the other two equal - length sides perpendicular to the building be $y$. Then the amount of fencing used is $x + 2y=500$, so $x = 500 - 2y$.

Step2: Express the area function

The area $A$ of the rectangle is $A=xy$. Substitute $x = 500 - 2y$ into the area formula, we get $A(y)=(500 - 2y)y=500y-2y^{2}$.

Step3: Find the derivative of the area function

Differentiate $A(y)$ with respect to $y$. $A^\prime(y)=\frac{d}{dy}(500y - 2y^{2})=500 - 4y$.

Step4: Find the critical points

Set $A^\prime(y)=0$. Then $500 - 4y = 0$. Solving for $y$ gives $4y=500$, so $y = 125$.

Step5: Find the value of $x$

Substitute $y = 125$ into $x = 500 - 2y$. Then $x=500-2\times125 = 250$.

Answer:

250, 125