You have been hired by a farmer to design a fenced-in rectangular enclosure for emus. The emus will require 720 square feet of area in which to roam, and the fence will cost 20 dollars per foot. The rectangular area will adjoin an existing wall, so a fence is only needed on three sides. You should minimize the cost of materials by choosing the dimensions carefully. What should the dimensions be, how much will it cost, and why? As you are writing the report for your client, you realize that the emus may try to bite through the fence. You may need a stronger fence that costs 30 dollars per foot instead. Also, although emus cannot ﬂy, they can jump, so you may need to cover the top of the enclosure with a wire mesh to prevent their escape. The wire mesh costs 2 dollars per square foot. Would these upgrades aﬀect the optimum dimensions? If so, how and why? If not, why not? How would these upgrades aﬀect the cost? In case the client is unable or unwilling to pay for the rectangular enclosure that you are designing, you should also propose an alternative shape for the enclosure. (Some possibilities: more sides, fewer sides, diﬀerent angles, a curved fence.) Make at least one proposal that costs less than the rectangular one. How much less will it cost, and why?

Question

You have been hired by a farmer to design a fenced-in rectangular enclosure for emus. The emus will require 720 square feet of area in which to roam, and the fence will cost 20 dollars per foot. The rectangular area will adjoin an existing wall, so a fence is only needed on three sides. You should minimize the cost of materials by choosing the dimensions carefully. What should the dimensions be, how much will it cost, and why? As you are writing the report for your client, you realize that the emus may try to bite through the fence. You may need a stronger fence that costs 30 dollars per foot instead. Also, although emus cannot ﬂy, they can jump, so you may need to cover the top of the enclosure with a wire mesh to prevent their escape. The wire mesh costs 2 dollars per square foot. Would these upgrades aﬀect the optimum dimensions? If so, how and why? If not, why not? How would these upgrades aﬀect the cost? In case the client is unable or unwilling to pay for the rectangular enclosure that you are designing, you should also propose an alternative shape for the enclosure. (Some possibilities: more sides, fewer sides, diﬀerent angles, a curved fence.) Make at least one proposal that costs less than the rectangular one. How much less will it cost, and why?

Answer 1

If the pen is x by y, then we have

xy=720
p = 2x+y = 2x+720/x
dp/dx = 2 - 720/x^2
x^2=360
x = 6√10
So, the pen is 6√10 by 12√10
As always, the area is max when the fence is divided equally among lengths and widths.

clearly the cost per foot of the fence does not affect the optimal dimensions.

If the fence costs $m/ft and the mesh costs $n/ft^2, then the cost is

c = m(2x+y)+720n
= 2mx + 720m/x + 720n
Clearly, neither the cost of the fence nor the cost of the mesh affects the dimensions

I suggest you explore a semicircular enclosure for lower cost.

Answer 2

To design the fenced-in rectangular enclosure for the emus, we need to determine the dimensions that will minimize the cost of materials.

Let's start by finding the dimensions of the rectangular enclosure that will provide the required area of 720 square feet. Since the rectangular area will adjoin an existing wall, we only need to consider three sides for the fence.

Let's assume the length of the rectangular enclosure is L, and the width is W. The area of the rectangle is given by the formula A = L * W = 720 square feet.

To minimize the cost of materials, we need to minimize the perimeter of the fence. The perimeter is given by P = L + 2W (since one side is already bordered by a wall).

To find the minimum cost, we need to express the cost function in terms of one variable. We can use the equation for the fence perimeter and substitute for one of the variables.

First, let's solve the area equation for one variable. We can solve for L in terms of W:

L = 720 / W

Substituting this value of L into the perimeter equation:

P = (720 / W) + 2W

To find the minimum cost, we can take the derivative of P with respect to W and set it equal to zero:

dP/dW = (-720 / W^2) + 2 = 0

Solving this equation, we find:

720 / W^2 = 2

W^2 = 720 / 2

W^2 = 360

W = sqrt(360) ≈ 18.97 feet

Substituting this value of W back into the original area equation:

L = 720 / W ≈ 720 / 18.97 ≈ 37.93 feet

The dimensions that will minimize the cost of materials for the rectangular enclosure are approximately 37.93 feet by 18.97 feet.

Now, let's calculate the cost. We have three sides of the fence to consider:

Cost of the longer side (L) = 30 dollars per foot * 37.93 feet
Cost of the two shorter sides (W) = 20 dollars per foot * 18.97 feet * 2

Total cost = 30 * 37.93 + 20 * 18.97 * 2 dollars

Now, let's evaluate whether the upgrades will affect the optimum dimensions and the cost.

If we need a stronger fence at 30 dollars per foot, the cost will increase. The dimensions may or may not change, depending on the specific requirements of the stronger fence. If the stronger fence requires wider supports or additional reinforcing, it may affect the width (W) dimension, but the length (L) might stay the same.

Similarly, if we need to cover the top of the enclosure with wire mesh at 2 dollars per square foot, it will add to the cost. The dimensions of the rectangle will remain the same.

Therefore, the upgrades may affect the cost but may or may not affect the optimum dimensions, depending on the specific requirements of the upgrades.

If the client is unable or unwilling to pay for the rectangular enclosure, we can propose an alternative shape that costs less. Here's one such proposal:

Instead of a rectangular enclosure, we can consider a circular enclosure. A circular fence may be a cost-effective alternative. To find the dimensions of the circular enclosure, we can use the formula for the area of a circle:

A = π * r^2

Given that the area A = 720 square feet, we can solve for the radius r:

720 = π * r^2

r^2 = 720 / π

r = sqrt(720 / π) ≈ 15.18 feet

The radius of the circular enclosure that will provide the required area of 720 square feet is approximately 15.18 feet.

To calculate the cost, we need to consider the circumference of the circle, which will be the cost of the fence:

C = 2 * π * r

Cost of the circular enclosure = 20 dollars per foot * C

Now, let's compare the costs between the rectangular and circular enclosures:

Cost of the rectangular enclosure = 30 * 37.93 + 20 * 18.97 * 2 dollars

Cost of the circular enclosure = 20 dollars per foot * 2 * π * 15.18 feet

The cost of the circular enclosure is likely to be less than the rectangular enclosure due to the lower perimeter of the circular fence. However, the specific cost difference will depend on the values of π and the cost per foot of the circular fence.

In conclusion, the dimensions for the rectangular enclosure that minimizes the cost of materials are approximately 37.93 feet by 18.97 feet. The total cost depends on the cost per foot of the fence and whether upgrades like a stronger fence or wire mesh are required. A circular enclosure can be a cost-effective alternative, with a radius of approximately 15.18 feet. The cost of the circular enclosure is likely to be less than the rectangular enclosure, but the exact difference depends on the cost per foot and the value of π.