# Calculus

A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100m^2. what are the dimensions of each pen that minimize the amount of fence that must be used?

1. 👍
2. 👎
3. 👁
1. If each pen has width x and length y (against the barn),
the area is 4xy, and the fence used is 5x+8y

So, each pen has area 100.

p = 5x + 8(100/x) = 5x + 800/x
dp/dx = 5 - 800/x^2

dp/dx = 0 when x = √160 = 4√10

x = 4√10
y = 25/√10

1. 👍
2. 👎
2. Now, if the barn is used as one wall of the pen, meaning only 3 sides have to be fenced, then

p = 5x+4y = 5x + 4(100/x) = 5x + 400/x
p' = 5 - 400/x^2

p' = 0 at x = 4√5
y = 25/√5

1. 👍
2. 👎
3. The equation for the perimeter of the adjacent pens is incorrect. They are being built against the barn meaning that the farmer does not use fencing against the barn. The correct equation to model that situation would thus be 5x+4y.

1. 👍
2. 👎

## Similar Questions

1. ### math

A shopkeeper bought pens at a rate of 12 pens for 150 rs and sold them at rate of 10 pens for 130rs.The number of pens he should sell to earn a net profit of 20rs is?

2. ### Calc 1

A farmer with 700 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?

3. ### math

Farmer Hodges has 50 feet of fencing to make a rectangular hog pen beside a very large barn. He needs to fence only three sides because the barn will form the fourth side. Studies have shown that under those conditions the side

4. ### Pre-Calc

How do you do this? A farmer has 336 feet of fencing and wants to build two identical pens for his prize-winning pigs. The pens will be arranged as shown. Determine the dimensions of a pen that will maximize its area.

1. ### Algebra

You plan to build four identical rectangular sheep pens in a row. Each adjacent pair of pens share a fence between them. You have a total of 368 feet of fence to use. Find the dimension of each pen such that you can enclose the

2. ### Math

A farmer has 80 feet of fencing, which she plans to use to fence in a plot of land for a pigpen. If she chooses to enclose a plot along the broad side of her barn, what is the largest area that can be enclosed? (Note: The side

3. ### Math

A rancher wants to fence in an area of 2000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?

4. ### Calculus

A rancher wants to construct two identical rectangular corrals using 300 ft of fencing. The rancher decides to build them adjacent to each other, so they share fencing on one side. What dimensions should the rancher use to

1. ### Calculus

A cattle rancher wants to enclose a rectangular area and then divide it into six pens with fencing parallel to one side of the rectangle (see the figure below). There are 480 feet of fencing available to complete the job. What is

2. ### College Calculus

A light is on the ground 20m from a barn. A 2m tall llama walks from the light directly toward the side of the barn at 1m/s. How fast is the height of the llama's shadow on the barn changing when he's 14m from the barn? (draw and

3. ### Calculus

A rancher wants to construct two identical rectangular corrals using 300 feet of fencing. The Ranger decides to build them adjacent to each other, so they share something on one side. What dimension to the range are used to

4. ### math

A farmer plans to enclose a rectangular region, using part of his barn for one side and fencing for the other three sides. If the side parallel to the barn is to be twice the length of an adjacent side, and the area of the region