Here's the question: A fence is to be built to enclose a rectangular area of 200 square feet. The fence along three sides is to be made of material that costs 5 dollars per foot, and the material for the fourth side costs 16 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Obviously it's an optimization problem, but I'm having trouble understanding how to go about doing this. What confuses me the most is the difference in price for specific fences. Any help would be appreciated.
$\endgroup$3 Answers
$\begingroup$Let $L$ be the length and $W$ be the width of the field. The required area gives you a relation between $L$ and $W$. Let the length be the direction that has the cheap fences on both sides (because you guess it will be longer, but if it comes out the other way that is OK.) What is the cost of the fence? Use the relation from the area to get cost as a function of (say) width. Differentiate, set to zero.....
$\endgroup$ $\begingroup$- Draw the rectangle.
- Label the two dimensions $x$ and $y$ or $l$ and $w$ or something.
- Use these two variables and the cost per foot values to construct a formula for total cost.
- Write down the equation for area (including the value of 200) in terms of the two variables. Then rearrange the formula to isolate one of the variables by itself on one side(solve for x or y).
- Plug that expression back into the cost formula and now you have a cost equation with just one variable that you can optimize.
If that doesn't make sense, let me know which step you get lost on and I can be more specific.
$\endgroup$ $\begingroup$Hint: Let $x$ be one side of the rectangle and $y$ the other. Furthermore, assume that the fourth side made of the more expensive material is used on one of the "$y$" sides of the fence*. Then the total cost is
$$5(2x + y) + 16(y)$$
And we want to minimize this under the constraint $xy = 200$.
*Note that we can do this "without loss of generality" (WLOG) to the problem. That is, we could just assume that the fourth side is one of the "$x$" sides and the problem would be the same.
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