Question

A farmer wants to build a rectangular pen and then divide it with two interior fences. The total area inside of the pen will be 264 square meters. The exterior fencing costs $15.60 per meter and the interior fencing costs $13.00 per meter. Find the dimensions of the pen that will minimize the cost.

Answer 1

To minimize the cost, the dimensions of the rectangular pen that will result in the least amount of exterior and interior fencing required are 22 meters by 12 meters. The total cost can be calculated using the formula: Total cost = (2L + 2W) x $15.60 + 2L x $13.00, where L is the length and W is the width of the pen.

Step-by-step explanation:

To minimize the cost, let's calculate the dimensions of the rectangular pen that will result in the least amount of exterior and interior fencing required.

Let's assume the length of the rectangular pen is L and the width is W.

The area of the rectangular pen is given as L * W = 264.

Since we have two interior fences dividing the pen, we can express the length and width as:

• L = 2W
• W = L/2

Substituting the second equation into the first equation, we get:

L = 2(L/2)

L = L

So regardless of the value we choose for L, it will always be equal to itself.

Therefore, we have an infinite number of solutions for L and W as long as they satisfy the equation L * W = 264.

For example, if we choose L = 22 meters, then W = 12 meters (L/2), and the dimensions of the rectangular pen that will minimize the cost are 22 meters by 12 meters.

To find the total cost, we can calculate the cost of the exterior fencing and the cost of the interior fencing:

Cost of exterior fencing = perimeter of the rectangular pen x cost per meter of exterior fencing

Cost of interior fencing = 2 x length of the pen x cost per meter of interior fencing

Substituting the values, the total cost would be:

Total cost = (2L + 2W) x $15.60 + 2L x $13.00

Plugging in the values L = 22 and W = 12, we can calculate the total cost.

Answer 2

x = 12 m and y = 22 m

Step-by-step explanation:

Total area = 264
`m^2`

∴ xy = 264

`$y=(264)/(x)$` ............(1)

Cost function =
`C(x,y) = 2 x (15.60) + 2y(15.60) + 2x(13)`

`C(x,y) = 57.2 x + 31.2y`

Therefore, using (1),

`$C(x) = 57.2x+31.2 \left((264)/(x) \right)$`

`$C(x) = 57.2x+(8236.8)/(x) \right)$`

So, cost C(x) minimum where C'(x) = 0

`$C'(x) = 57.2 - (8236.8)/(x^2)=0$`

`$x^2=(8236.8)/(57.2)$`

`$x^2=144$`

`$x=12$` m

Therefore,
`$y=(264)/(x)$`

`$=(264)/(12)$`

= 22 m

So the dimensions are x = 12 m and y = 22 m.