Questions: Christian 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 Christian has 750 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 an function for the area A of the enclosure in terms of w. A(w)= b) What width w would maximize the area? w= c) What is the maximum area? A= c) What is the feet

Christian 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 Christian has 750 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 an function for the area A of the enclosure in terms of w.

A(w)=

b) What width w would maximize the area?

w=

c) What is the maximum area?

A= c) What is the feet

Transcript text: Christian 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 Christian has 750 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 an function for the area $A$ of the enclosure in terms of $w$. \[ A(w)= \] b) What width $w$ would maximize the area? \[ w= \] c) What is the maximum area? \[ A=\square \text { c) What is the feet } \]

Solution

Solution Steps

To solve the problem of maximizing the area of a rectangular enclosure with one side against a barn, we can follow these steps:

Define Variables: Let $ w $ be the width of the enclosure (perpendicular to the barn) and $ l $ be the length of the enclosure (parallel to the barn).
Express Length in Terms of Width: Since only three sides need fencing, the total length of the fencing used is $ l + 2w = 750 $. Solve for $ l $ in terms of $ w $.
Area Function: Write the area $ A $ of the enclosure as a function of $ w $.
Maximize the Area: Use calculus (find the derivative and set it to zero) to find the width $ w $ that maximizes the area.
Calculate Maximum Area: Substitute the optimal width back into the area function to find the maximum area.

Step 1: Define the Variables

Let $ w $ be the width of the enclosure (perpendicular to the barn) and $ l $ be the length of the enclosure (parallel to the barn). The total amount of fencing available is 750 feet.

Step 2: Express Length in Terms of Width

Since one side of the enclosure is against the barn, the fencing requirement can be expressed as: \[ l + 2w = 750 \] From this, we can solve for $ l $: \[ l = 750 - 2w \]

Step 3: Write the Area Function

The area $ A $ of the enclosure can be expressed as: \[ A(w) = w \cdot l = w \cdot (750 - 2w) = 750w - 2w^2 \]

Step 4: Maximize the Area

To find the width $ w $ that maximizes the area, we take the derivative of the area function and set it to zero: \[ \frac{dA}{dw} = 750 - 4w \] Setting the derivative equal to zero gives: \[ 750 - 4w = 0 \implies 4w = 750 \implies w = \frac{750}{4} = 187.5 \]

Step 5: Calculate the Maximum Area

Substituting $ w = 187.5 $ back into the area function to find the maximum area: \[ A(187.5) = 750(187.5) - 2(187.5)^2 = 140625 - 7031.25 = 133593.75 \]