The length of a rectangle is 2 ft more than twice the width. The area is 60 ft^2. Find the length and width of the rectangle.

Question

The length of a rectangle is 2 ft more than twice the width. The area is 60 ft². Find the length and width of the rectangle.

Answer 1

Here, this question is about extracting information from the text: the only thing you need to know in advance is the formula for area of a rectangle which is length times width equals area ( l x w = a).

To unpack the words, here is a list of what we are told:

1. The length is given to us as a function of width which means if I’m selecting something to be “x,” my variable I’m going to make it the width. Thus first things first - “x” will equal my width. Remember though that when we solve this equation we are building it will give us the LENGTH
2. We are told “2 feet more than.” “More than sounds a lot like addition doesn’t it? Therefore somewhere in my equation I’m adding two to something!
3. We are told “Twice the width.” Twice sounds a lot like multiplication doesn’t it? Therefore, somewhere in my equation I’m multiplying something by 2!
4. Finally we are told that the area is 60 square feet! This means what? That l x w = 60! This means that whatever I do in my equation, after the equals sign will be “60l”
5. Okay! Time to build the equation!
6. First… it’s equal to 60 so it begins with … = 60
7. Next, the first thing we said is the width would be x so since we are using l x w = a, I can put in x for the width so… l * x = 60
8. Now the tricky part…how is the length defined? Well we know! It’s 2 times the width plus 2! So…this is what the whole thing looks like: (2x+2)x = 60
9. Now we solve:
10. Distribute the x to eliminate the parentheses: 3x + 2x = 60
11. Combine like terms: 5x = 60
12. Divid both sides by 5: x = 12
13. REMEMBER THAT X IS ONLY THE LENGTH!
14. So the width is 12 minus two divided by 2 or 5
15. So Length = 12, Width = 5
16. check it: l x w = A, 12 x 5 = 60

Done

Answer 2

With problems like these where they tell you how to find something in terms of something else, I like to pick a number (in this case for the width) and see what the length would be in that case.

Let's say the width is 10 feet (again this is a random number). The problem says that the length is "2 feet more than twice the width". Well, twice the width would be 20 feet. 2 feet more than that would be 22. Using numbers can sometimes help you write the mathematical expression easier. I found 22=2(10)+2. Here 22 was the length L and 10 was the random width, W, I wanted to try out. So the expression for finding the length would be L=2W+2.

Now we want to find the area of the rectangle. The Area of a Rectangle is found by the formula A=LW. We are told the Area is 60 square feet, so 60=LW. The trick here is there are TWO unknowns, so to make it so we have only 1 unknown, we can replace L, with the expression we found earlier: L=2W+2.

This makes the equation 60=(2W+2)(W). This looks more complicated, but now there is only one variable, so we can solve for it using algebra.

First step, lets distribute the W: 60=2W^2+2W

So this equation becomes a quadratic equation. There are a LOT of ways to solve quadratic equations, some faster than others, but let's stick with the quadratic formula since we KNOW it will work, but this one is easily factored as well.

To use the quadratic formula, we need one side set to 0, so we subtract 60 from both sides

2W^2+2W-60=0. I notice each term is divisible by 2 so lets just divide that out to make the numbers smaller

W^2+2W-30=0

Now that it is in the correct format, we identify the coefficients, a, b, and c as 1, 1, and -30 respectively.

plugging into the formula we have

W=[-1 +/- sqrt{1^2-4(1)(-30)}]/2(1) which is x=5 and x=-6 when you evaluate it.

This is a real world problem in which a negative value for W doesn't work, so we know that W=5.

Since L=2(5)+2, we know the length is 12

Check our work 60=5*12 has required.