This may seem a bit trickier, and that's probably because it is. Since we need to use both area and perimeter here, we should know that *A* = *lw* and *P* = 2*l* + 2*w*. Here, we have two equations and two unknowns, so we should be able to solve them both. And you thought algebra wouldn't come in handy. First, we can substitute the perimeter into the formula and solve for *l* in terms of *w*. *P* = 2*l* + 2*w* 120 = 2*l* + 2*w* 2*l* = 120 – 2*w*
*l* = 60 – *w*
Now, we can make use of the area formula and substitute 60 – *w* for *l*. Then, we can multiply them out and solve for *w*. *A* = *lw* 800 = (60 – *w*)*w* 800 = 60*w* – *w*^{2}
*w*^{2} – 60*w* + 800 = 0
That looks like a quadratic equation, and indeed it is. Let's use whatever algebraic methods we remember in order to solve it. We'll factor, but you don't have to. (We understand. It's been a while.) Factoring means we need two numbers that add to make -60 and multiply to make 800. How about -20 and -40? (*w* – 20)(*w* – 40) = 0 That means *w* = 20 and *w* = 40. How do we know which one is right? Let's compare it to the length to find out. Remember how we solved for *l* in terms of *w*? Well, if *w* = 20, then we have *l* = 60 – 20 = 40. If *w* = 40, then *l* = 60 – 40 = 20. Basically, whatever direction we choose to spin it, our side lengths are 20 cm and 40 cm. If you're a real stickler for rules, *w* = 20 cm and *l* = 40 cm (because length is longer), but those are the dimensions of the rectangle. |