The length of a rectangle is 26 centimeters more than its width. Given that its area is 120 centimeters squared, determine its perimeter.
So, we have a rectangle. And we’ll call its width 𝑤. We don’t know the length or the width. But we do know some information about the length. The length equals 26 centimeters more than the width. We could write that as 𝑤 plus 26. We also know that its area equals 120 centimeters squared. We know that the area of a rectangle is length times the width. The area equals 120. The length equals the width plus 26. And the width equals the width.
We can distribute the width, multiplying 𝑤 times 𝑤 gives us 𝑤 squared. And 𝑤 times 26 equals 26𝑤. Our new equation says 120 equals 𝑤 squared plus 26𝑤. To find the value of 𝑤, we can treat this like an equation. And we can solve by subtracting 120 from both sides. Zero equals 𝑤 squared plus 26𝑤 minus 120.
To find the value of 𝑤, we’ll factor this equation. The first term of each expression would be 𝑤. We need two factors that multiply together to equal negative 120 and when added together equals positive 26. We have negative one times 120, positive one times negative 120, negative two times 60, positive two times negative 60, negative three times 40, three times negative 40, negative four times 30, four times negative 30. We could keep going, but remember the goal is to find two factors that add together to equal positive 26. Negative four plus 30 equals 26.
And that means we can substitute negative four and positive 30 into our expressions. Remember, our end goal is to find the perimeter. But in order to find the perimeter, we need to know the width and the length. We’re almost to finding the width. We take each of these terms and set them equal to zero. 𝑤 minus four equals zero. Add four to both sides. And we see that 𝑤 equals four. 𝑤 plus 30 equals zero. So, we subtract 30 from both sides. And we get that 𝑤 equals negative 30.
We know that we’re talking about length here. And that means 𝑤 equals negative 30 is not a possible solution. The only possible solution is 𝑤 equals four. Which means the width of this rectangle equals four centimeters. And that means the length equals four plus 26. The length equals 30 centimeters.
The perimeter of a rectangle is two times the length plus the width. For us, that’s two times 30 plus four. Two times 34 equals 68. Remember that perimeter is the distance around the shape. And that distance is a measure of length. So, our units is centimeters, not centimeters squared. The distance around this rectangle is 68 centimeters.