Problem: Billy Bob is building a rectangular pen for his beloved Komodo dragon, which he fondly refers to as Bessie. He knows that her pen needs to be 14 feet longer than it is wide. Also, Billy Bob has 92 feet of fencing. How long and how wide should Bessie's pen be?

Now, make no mistake, Billy Bob is a bright, well-liked fellow. But he doesn't remember any any algebra, so he needs a little help figuring out the dimensions (length and width) of the pen. Explain how to do it using the techniques you have learned in this course.

Solution: Billy Bob, you need to translate the information the problem gives us into an equation whose solution answers the question.

First, we are going to assign variables to the quantities we want to find, namely, the length and the width of the pen which we will denote as $\ell$ and $w$, respectively.

Second, we are going to translate the problem language into a mathematical statement which relates both $\ell$ and $w$. We know that "the pen needs to be 14 feet longer than it is wide." That is, the length is 14 feet longer than the width. The translation of the above is as follows: $$ \begin{array}{ccc} \mbox{The length} & \mbox{is} & \mbox{14 feet longer than the width.} \\ \ell & = & w+14 \end{array} $$ So, we know $\ell=w+14.$

Third, we are now going to draw a picture of the rectangular pen which will help us to better understand the problem. A generic rectangle of of length $\ell$ and width $w$ looks like:

	$\ell$
$w$		$w$
	$\ell$

Using the above information, that is, $\ell=w+14$, the above rectangle may be redrawn as:

	$w+14$
$w$		$w$
	$w+14$

You can see that the above is a good visual representation of the problem. We can now form an equation: since the perimeter is how far we would have to walk around the pen, starting at the lower left corner, walking clockwise, the distance we walk may be expressed as $$ \begin{array}{ccccc} \mbox{perimeter} &= \mbox{lower left to upper left} & +\mbox{upper left to upper right} & +\mbox{upper right to lower right} & +\mbox{lower right to lower left} \\ P &=w &+(w+14) &+w &+(w+14) \end{array} $$ More succinctly, $P=w+(w+14)+w+(w+14).$ Finally, we know that we have 92 feet of fencing; this tells us that the perimeter $P$ is 92 feet. This fact combined with the above paragraph gives us the following equation: $$92=w+(w+14)+w+(w+14).$$ We now solve this equation: $$ \begin{array}{rclr} 92&=&w+(w+14)+w+(w+14) & \mbox{Original Equation}\\ w+(w+14)+w+(w+14)&=&92 & \mbox{Rewrite with variable on left side.}\\ 4w+28&=&92 & \mbox{Combine like terms}\\ 4w+28-28&=&92-28 & \mbox{Subtract 28 from both sides.}\\ 4w&=&64 & \mbox{Simplify.}\\ \frac{4w}{4}&=&\frac{64}{4} & \mbox{Divide both sides by 4.}\\ w&=&16 & \mbox{Simplify.}\\ \end{array} $$ Thus, the width is $w=16$ feet. Since the length is 14 feet more than the width, we have that the length is $\ell=w+14=16+14=30$ feet.

So, the pen will be 16 feet by 30 feet.

Now, we should ask ourselves if these numbers are reasonable. They do seem reasonable. So let's check the numbers themselves to convince ourselves that we really nailed it. We know that that the pen needs to be 14 feet longer than it is wide; the length of 30 feet is certainly 14 feet longer than the width of 16 feet. But we also know that the perimeter must be 92 feet. That is $30+16+30+16$ should add up to 92. A quick check reveals that $$ \begin{array}{ll} & 30+16+30+16\\ =&46+46\\ =&92 \end{array} $$ So our answer checks.

Therefore, the answer to our problem of the week is:

Billy Bob needs to build Bessie a pen which is 16 feet by 30 feet.

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