Question

The length of a rectangle is 5 feet more than 4 times the width. If the area is 84 square feet, find the width and the length.

Answer 1

So let's just say length = L and width = W. It says the length is 5 more than 4 times the width. So translating into symbols. 5 more means + 5. And 4 times widthmeans 4w. So 5 more than 4 times thewidth is the same as 5 + 4w. So we can say that thelength is the exact same thing as 5 + 4w. Now it wants us to use this information plus the fact that the area is 84 to find W and L.

Well, area of a rectangle is LW. So in this problem, LW = 84 But we justsaid that L is the same as 5 + 4w. Because of that, I can replace L and write (5+4w)w = 84. Now we multiply this out and get 4w^2 + 5w = 84. This is starting to look like a quadratic, soO bethca we can treat it like one and factor it to solve for x. So to do that, we set thewhole equation equal to 0 by moving 84 to the other side, subtracting it from both sides
4w^2 + 5w - 84 = 0
Now in order to factor this, we multiply 4 and -84, then see if we can find two factors of that answer that add up to 5. So first off, 4 times -84= -336. Now to see if we can factor 336 to come up with 2 factors that add up to 5



So I find 21 and 16. If I make the 16 negative and the21 positive, Ill have satisfied the conditions im looking for. So now that 5w part of the quadratic will be replaced with -16w and 21w, giving us
4w^2 - 16w + 21w - 84

At this point, we must factor by grouping. So we wantto make two groups of 2 and we want those groups to be logical. Because 4 and -16 are both multiples of 4, those two are logical to group, meaning ill also put the 21 and -84 ina group
(4w^2 - 16w) + (21w-84)
Now we want tofactor out the largest possible number and variable from each group. In the left grouping, we can factor out a 4w as our largest factor. Doing that givesus
4w(w-4)
As forthe right grouping, a 21 can befactored out and nothing more. Doign thatgives
21(w-4)
We know we did something right because both times we factoredwe were left with w-4. If these were not the same we know we did somethign wrong. So finally, the factored form takes oneof the w-4) factors aswell as the two terms we factored out, meanign the full factored answer is (w-4)(4w+21)
Now to actually solve for w, we need to take each factored group and set it equal to 0, meanign we have

w -4 = 0
4w + 21 = 0 solving for both, I get

w = 4 and w = -21/4

Well, a width HAS to be positive. You cant havea negative length or height. This means the width must be 4. So knowing that the width is 4, we were told way back that the length is 5 + 4w. Andsince w is 4, we can say L = 5 + 4(4) and L = 21. So the answersforthis problem is L = 21 and W= 4. And you can even check it because 21 times 4 equals the area of 84 like we want.