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? \[ A=\square \text { c) What is the feet } \]
failed

Solution

failed
failed

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:

  1. 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).
  2. 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 \( w \).
  3. Area Function: Write the area \( A \) of the enclosure as a function of \( w \).
  4. Maximize the Area: Use calculus (find the derivative and set it to zero) to find the width \( w \) that maximizes the area.
  5. 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: \[ A(187.5) = 750(187.5) - 2(187.5)^2 = 140625 - 7031.25 = 133593.75 \]

Final Answer

The width \( w \) that maximizes the area is \( \boxed{w = 187.5} \) ft, and the maximum area is \( \boxed{A = 133593.75} \) square feet.

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
find the mean Score Percent of final grade Homework 88 10 Quiz 80 10 Quiz 93 10 Project 98 35 Final Exam 86 35
failedAnswered
Find the length of RX
failedAnswered
Question 4 1 pts Which of these documents is more commonly written in business? Essay Report Question 5 1 pts Which of these documents uses first-person ("I," "me," "you")? Essay Report Question 6 1 pts Which of these documents includes an Executive Summary? Essay Report
failedAnswered
A tank contains 80 kg of salt and 1000 L of water. A solution of a concentration 0.04 kg of salt per liter enters a tank at the rate 6 L / min. The solution is mixed and drains from the tank at the same rate. (a) What is the concentration of our solution in the tank initially? concentration = 0.08 (kg / L) (b) Set up an initial value problem for the quantity y, in kg, of salt in the tank at time t minutes. dy/dt = (kg / min) y(0) = (kg) (c) Solve the initial value problem in part (b). y(t) = (d) Find the amount of salt in the tank after 3 hours. amount = (kg) (e) Find the concentration of salt in the solution in the tank as time approaches infinity. concentration = (kg / L)
failedAnswered
As the CEO of a textile mill, you understand that a devastating fire, even when your company is well insured, could result in property damage, loss of business, increased insurance premiums, and a negative public image. Therefore, it is in your best interest to create effective fire prevention practices. This type of thinking among the business community has resulted in: - increased government regulation. - successes in protecting property and saving lives. - new options in insurance. - security deficiencies in the private sector.