Solve for Width and Length of a Rectangular Barn Wall with Given Area and Length-Width Relationship

Question

The area of a rectangular wall of a barn is 70 square feet. Its length is 4 feet longer than twice its width. Find the length and width of the wall of the barn. (First part: find the width)

Accepted Answer

width = 5 feet; length = 14 feet

Answer

Step 1: Define variables. Let the width of the rectangular wall be $ w $ feet. According to the problem, the length is 4 feet longer than twice the width, so length $ l = 2w + 4 $ feet. Step 2: Use the area formula for a rectangle. The area $ A = l \times w $, and we know the area is 70 square feet. Substitute the expressions for $ l $ and $ A $ into the formula: $ 70 = (2w + 4)w $. Step 3: Expand and rearrange the equation into standard quadratic form. Expand the right side: $ 2w^2 + 4w = 70 $, then subtract 70 from both sides to get $ 2w^2 + 4w - 70 = 0 $. We can simplify this by dividing all terms by 2: $ w^2 + 2w - 35 = 0 $. Step 4: Factor the quadratic equation. We need two numbers that multiply to -35 and add to 2. These numbers are 7 and -5, so the equation factors to $ (w + 7)(w - 5) = 0 $. Step 5: Solve for $ w $. Set each factor equal to 0: $ w + 7 = 0 $ gives $ w = -7 $, and $ w - 5 = 0 $ gives $ w = 5 $. Since width cannot be a negative number, we discard $ w = -7 $. Step 6: Calculate the length. Substitute $ w = 5 $ into the length formula: $ l = 2(5) + 4 = 10 + 4 = 14 $ feet.