Question

The perimeter of a rectangle is 16 inches. The equation that represents the perimeter of the rectangle is 2l + 2w = 16, where l represents the length of the rectangle and w represents the width of the rectangle. Which value is possible for the length of the rectangle?

Answer 1

Assuming a width of 2 inches, a possible value for the length of a rectangle with a perimeter of 16 inches is 6 inches.

Step-by-step explanation:

The student has provided the perimeter equation for a rectangle, 2l + 2w = 16, and is looking for a possible value for the length of the rectangle. To find a possible length, we can assume a value for w, the width, and calculate the corresponding length. Let's say the width is 2 inches. Substituting w = 2 into the equation we get 2l + 2(2) = 16, which simplifies to 2l + 4 = 16. Subtracting 4 from both sides, we get 2l = 12 and dividing both sides by 2, we find l = 6. Therefore, a possible value for the length of the rectangle is 6 inches.

Answer 2

To solve this equation, we remember that perimeter p=16 in, and p=2L+2W. Replacing for p and using the distributive property in reverse, we can let our equation look like 16=2(L+W). Divide both sides by 2 to isolate the variables; our equation is now 8=L+W. From here, L, or the value of the rectangle's length, may be any value between 1 and 7, provided that the width W sums with it to make 8. For example, L could be 5 and W 3, and all would work out; any combination of 1-7, 2-6, 3-5, and 4-4 (a square) are possible combinations of the rectangle's side lengths. Obviously, a side length for a rectangle cannot be negative and cannot be 0, which is why our set of solutions is so small. For greater confidence, one can go back to the original equation and test the solution by plugging values into the formula. For example, 16=2(5)+(2)3 --> 16=10+6 --> 16=16. All good! If you're being really technical about the definitions of length vs. width, then you might need to stipulate that the length should always be the longer side, meaning that your set of results for L would only be {4,5,6,7}. A square is a rectangle, so the 4 value should be valid.