question 18 aidan wants to build a rectangular enclosure for his animals. one side of the pen will be…

question 18 aidan wants to build a rectangular enclosure for his animals. one side of the pen will be against the barn, so he needs no fence on that side. the other three sides will be enclosed with wire fencing. if aidan has 700 feet of fencing, you can find the dimensions that maximize the area of the enclosure. a) let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). write a function for the area a of the enclosure in terms of w. (hint first write two equations with w and l and a. solve for l in one equation and substitute for l in the other). a(w)= b) what width w would maximize the area? w = ft c) what is the maximum area? a = square feet > next question

question 18 aidan wants to build a rectangular enclosure for his animals. one side of the pen will be against the barn, so he needs no fence on that side. the other three sides will be enclosed with wire fencing. if aidan has 700 feet of fencing, you can find the dimensions that maximize the area of the enclosure. a) let w be the width of the enclosure (perpendicular to the barn) and let l be the length of the enclosure (parallel to the barn). write a function for the area a of the enclosure in terms of w. (hint first write two equations with w and l and a. solve for l in one equation and substitute for l in the other). a(w)= b) what width w would maximize the area? w = ft c) what is the maximum area? a = square feet > next question

Answer

Explanation:

Step1: Set up the perimeter - related equation

Since one - side of the rectangle is against the barn and the total length of the fencing is 700 feet, we have $2w + l=700$, so $l = 700 - 2w$.

Step2: Write the area function

The area of a rectangle is $A=wl$. Substitute $l = 700 - 2w$ into the area formula, we get $A(w)=w(700 - 2w)=700w-2w^{2}$.

Step3: Find the width that maximizes the area

The function $A(w)=-2w^{2}+700w$ is a quadratic function in the form $y = ax^{2}+bx + c$ with $a=-2$, $b = 700$, and $c = 0$. The vertex of a quadratic function $y = ax^{2}+bx + c$ has its x - coordinate (in our case, the value of $w$ for the maximum of $A(w)$) given by $w=-\frac{b}{2a}$. Substitute $a=-2$ and $b = 700$ into the formula: $w=-\frac{700}{2\times(-2)}=\frac{700}{4}=175$ feet.

Step4: Find the maximum area

Substitute $w = 175$ into the area function $A(w)=700w-2w^{2}$. $A(175)=700\times175-2\times(175)^{2}=122500 - 2\times30625=122500-61250 = 61250$ square feet.

Answer:

a) $A(w)=700w - 2w^{2}$ b) $175$ c) $61250$