caden wants to build a rectangular pen for some farm animals. one side of the pen will be against a barn, so…

caden wants to build a rectangular pen for some farm animals. one side of the pen will be against a barn, so no fence is needed on that side. the other three sides will require fencing. if caden has 950 feet of fencing, find the dimensions that maximize the area of the pen, using all of the available fencing material. let w be the width of the enclosure and let l be the length of the enclosure (as shown above). write an function for the area a of the enclosure in terms of w. a(w)= what width w and length l would maximize the area? w= ft and l= feet what is the maximum area? a= square feet

caden wants to build a rectangular pen for some farm animals. one side of the pen will be against a barn, so no fence is needed on that side. the other three sides will require fencing. if caden has 950 feet of fencing, find the dimensions that maximize the area of the pen, using all of the available fencing material. let w be the width of the enclosure and let l be the length of the enclosure (as shown above). write an function for the area a of the enclosure in terms of w. a(w)= what width w and length l would maximize the area? w= ft and l= feet what is the maximum area? a= square feet

Answer

Explanation:

Step1: Relate length and width to total fencing

The total length of the three - sided fence is $2w + l=950$, so $l = 950 - 2w$.

Step2: Write the area function

The area of a rectangle is $A=wl$. Substitute $l = 950 - 2w$ into the area formula, we get $A(w)=w(950 - 2w)=950w-2w^{2}$.

Step3: Find the derivative of the area function

The derivative of $A(w)=950w - 2w^{2}$ with respect to $w$ is $A'(w)=950 - 4w$.

Step4: Set the derivative equal to zero to find critical points

Set $A'(w)=0$, then $950 - 4w = 0$. Solving for $w$ gives $4w=950$, so $w=\frac{950}{4}=237.5$.

Step5: Find the length

Substitute $w = 237.5$ into the equation $l = 950 - 2w$. Then $l=950-2\times237.5=950 - 475 = 475$.

Step6: Find the maximum area

Substitute $w = 237.5$ and $l = 475$ into the area formula $A = wl$. So $A=237.5\times475 = 112812.5$.

Answer:

$A(w)=950w - 2w^{2}$ $w = 237.5$ ft and $l = 475$ feet $A = 112812.5$ square feet