A certain rectangle's length is 8 feet longer than its width. If the area of the rectangle is 105 square feet what are its dimensions?

Answer 1

The rectangle is 15 feet long by 7 feet wide. But let's do the math so you come away with something other than just some numbers for an answer. Length (l) times width (w) will get us the area of a rectangle, as you know. These are the variables in the problem, and they can have different values (hence their being called variables). But we also have one of them expressed in terms of the other one. And we have the area. Let's take that to the "machine" we set up, which is the expression we will create (the "formula" if you prefer) that will lead us to the answer. Roll up your sleeves and let's do this. Arectangle = l x wl = w + 8 [we have l in terms of w, so we can put that back into the original expression] Arectangle = (w + 8) x w [put in the known area (105) and do the multiplication] 105 = w2 + 8w [we used the distributive property of multiplication, as you see, and now we subtract 105 from each side] w2 + 8w - 105 = 0 [yes, we have a second degree equation, but no panic as we'll factor] (w + 15) x (w - 7) = 0 [the two factors, when multiplied, give the original expression] Here's the deal. If we have two numbers that when multiplied together give us a product of zero, then either one of the numbers must be zero, or the othernumber must be zero, or both numbers must be zero. There are two answers for w here, so let's find both of them by setting each number equal to zero and then solving for it. w + 15 = 0 , w = -15 [we subtracted 15 from each side] w - 7 = 0 , w = 7 [we added 7 to each side] We were solving for the width of a rectangle. Can a rectangle have a width with negative length (-15) as solved? No, it can't. But it can have a width of 7 as solved. Let's take the 7 and plug it back into the original expression and solve for l(the length) there. Arectangle = l x w 105 = lx 7 l = 105 / 7 = 15 [105 divided by 7, which is the width, gives us the length] The length of the rectangle is 15 feet as discovered. The 7 x 15 = 105, and 7 + 8 = 15, so we've checked it and found it to be good. Piece of cake.