Let's get a feel for what information we need for this problem. We ultimately want to <em>find </em>the dimensions of the mural. What we <em>know </em>are
- The area of the mural (70 ft²)
- The relationship between length and width (the length is four less than twice the width)
Let's see how this information can help us through the different parts of the problem.
<h3>"Write the length in terms of width."</h3>
A good place to start is with naming our variables so we can start to put together equations. Let's call our length l and our width w. We're told that l is "four feet less than twice" w, or symbolically:
l = 2w - 4
Any time we have to use an l, we're now able to substitute the expression 2w - 4 instead.
<h3>"Write a function that represents the area in terms of width."</h3>
Again, let's start with giving our function a name. If we call the area of the mural A, we use the abbreviation A(w) to mean "A as a function of w," or "A of w" for short.
We know that the area A of a rectangle is normally just its length times its width - A = lw - but sense we want this function as a function of w alone, we can replace l with the expression 2w - 4 from earlier to give us
A(w) = (2w - 4)w = 2w² - 4w
<h3>"Find the dimensions."</h3>
We're finally ready to tackle this problem's main question. Our function from the last section tells us what the area of the mural is entirely in terms of w, and the first section tells us how to find l if we already know w. Going back to one of the first bullet points, we also know that <em>the area of the mural is 70 ft²</em>, so we can say that we want to know what w is when A(w) = 70, or equivalently, when
2w² - 4w = 70
To solve for w, we can first divide away a 2 on either side, leaving us with
w² - 2w = 35
or equivalently,
w(w - 2) = 35
The only value w can be to make this true is 7, since 7(7 - 2) = 7(5) = 35, so we know that the width must be 7 ft. The area is 70 ft², so the length of the mural must be 10 ft., since 7 x 10 = 70.
