# Flowerchild1993's question at Yahoo Answers regarding minimizing cost of fence

• MHB
• MarkFL
In summary, the cheapest way to enclose a rectangular field with an area of 770 ft squared is to have two opposite sides with a length of $\frac{4\sqrt{1155}}{3}$ feet and the other two sides with a length of $\frac{\sqrt{1155}}{2}$ feet. This will result in a cost of $3 per foot for two sides and$8 per foot for the other two sides.
MarkFL
Gold Member
MHB
Here is the question:

Calculus help! Word problem?

A rectangular field is to be enclosed on four sides with a fence. Fencing costs $3 per foot for two opposite sides, and$8 per foot for the other two sides. Find the dimensions of the field of area 770 ft squared that would be the cheapest to enclose.

Here is a link to the question:

Calculus help! Word problem? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

Hello flowerchild1993,

Let's let $x$ be the length of one pair of sides, for which fencing costs $C_x$ in dollars per foot and $0$ be the length of the other pair of sides, for which fencing costs $C_y$ in dollars per foot. All measures and constants are positive.

The cost function is therefore:

$$\displaystyle C(x,y)=2C_xx+2C_yy$$

Let the area of the field be $A$, and so we have:

$$\displaystyle A=xy\,\therefore\,y=\frac{A}{x}$$

and so we may write the cost function in terms of one variable $x$:

$$\displaystyle C(x)=2C_xx+2AC_yx^{-1}$$

Now, to find the extrema, we differentiate with respect to $x$ and equate to zero:

$$\displaystyle C'(x)=2C_x-2AC_yx^{-2}=0$$

Multiplying through by $$\displaystyle \frac{x^2}{2}$$ we may arrange this as:

$$\displaystyle x^2=\frac{AC_y}{C_x}$$

Taking the positive root, we find:

$$\displaystyle x=\sqrt{\frac{AC_y}{C_x}}$$

And so:

$$\displaystyle y=\frac{A}{\sqrt{\frac{AC_y}{C_x}}}=\sqrt{\frac{AC_x}{C_y}}$$

This is why I used variables to denote the constants of the problem...we can now see how the dimensions of the field will change with respect to changes made to any of the constants. We see that if $C_x$ increases then $x$ will decrease and $y$ will increase, etc. We also see that:

$$\displaystyle \frac{x}{y}=\frac{C_y}{C_x}$$

This ratio also tells us how the dimensions will vary based on the costs.

Using the second derivative test for relative extrema, we find that:

$$\displaystyle C''(x)=4AC_yx^{-3}>0$$ for $$\displaystyle x>0$$ which we have assumed, and so we know this critical value is at a minimum.

Now it's just a matter of using the given data to find the required dimensions. Let's let $x$ be the pair of sides with the cheaper fencing, so we have:

$$\displaystyle C_x=3,\,C_y=8,\,A=770$$

$$\displaystyle x=\sqrt{\frac{770\cdot8}{3}}=\frac{4\sqrt{1155}}{3}$$

$$\displaystyle y=\sqrt{\frac{770\cdot3}{8}}=\frac{\sqrt{1155}}{2}$$

To flowerchild1993 and any other guests viewing this topic, I invite and encourage you to post other optimization problems here in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.

## 1. How can I minimize the cost of building a fence?

To minimize the cost of building a fence, you can consider using cheaper materials such as wood or vinyl instead of more expensive options such as wrought iron. You can also reduce the size of the fence or opt for a simpler design to save on labor costs. Additionally, shopping around for different contractors and comparing quotes can help you find the best deal.

## 2. Is it better to build a fence myself or hire a professional?

This ultimately depends on your level of skill and the complexity of the fence design. If you have experience in building fences and access to the necessary tools, it may be more cost-effective to do it yourself. However, if you are not confident in your abilities, it may be worth hiring a professional to ensure the fence is built correctly and to save time and potential costly mistakes.

## 3. Are there any alternative options to building a traditional fence?

Yes, there are alternative options to building a traditional fence that may be more cost-effective. These include using plants as a natural barrier, using wire mesh fencing, or using a combination of both. These options may require less labor and materials, thus reducing the overall cost.

## 4. What factors should I consider when calculating the cost of a fence?

When calculating the cost of a fence, you should consider the materials, labor, and any additional features such as gates or decorative elements. You should also factor in the size and complexity of the fence design, as well as any potential obstacles or terrain that may affect the construction process.

## 5. Are there any tips for negotiating a lower price with a fencing contractor?

Yes, there are a few tips for negotiating a lower price with a fencing contractor. First, be sure to get multiple quotes and use them to negotiate a better deal. You can also offer to pay in cash or provide materials to reduce the cost. Additionally, be clear about your budget and any specific requirements you have to help the contractor find ways to lower the cost.

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