Question

Consider the collection of all rectangles that have lengths 2 in. less than twice their widths. Find the possible widths (in inches) of these rectangles if their perimeters are less than 200 in.

Answer 1

To find the possible widths of rectangles with a perimeter less than 200 inches, where the length is 2 inches less than twice the width, we use the perimeter formula and inequalities to determine that the widths are less than 34 inches.

Step-by-step explanation:

To solve for the possible widths of rectangles that have lengths 2 inches less than twice their widths with a perimeter less than 200 inches, we first need to formulate an equation based on the given conditions. We let the width of the rectangle be w inches. Therefore, the length would be 2w - 2 inches. The perimeter P of a rectangle is found by the formula P = 2l + 2w, where l is the length and w is the width.

Substituting the expressions for the length and width, we get:

P = 2(2w - 2) + 2w
P = 4w - 4 + 2w
P = 6w - 4

Since the perimeter is less than 200 inches:

6w - 4 < 200

Adding 4 to both sides gives us:

6w < 204

Dividing by 6:

w < 34

So, the possible widths of the rectangles are less than 34 inches.