Question 7, 1.5.23 HW Score: \( 96.43 \%, 13.5 \) of 14 points Part 2 of 2 Points: 0.5 of 1 Save The area of a rectangular wall of a barn is 36 square feet. Its length is 6 feet longer than twice its width. Find the length and width of the wall of the barn. width \( =3 \) feet length \( = \) \( \square \) feet

To solve this problem, we can set up a system of equations based on the information given. You know that the area (A) of the rectangle is given by the formula \( A = \text{length} \times \text{width} \), which in this case can be written as \( 36 = \text{length} \times \text{width} \). Since the length is said to be 6 feet longer than twice the width, we can express that as \( \text{length} = 2 \times \text{width} + 6 \). Substituting this back into the area equation, we get: \( 36 = (2 \times \text{width} + 6) \times \text{width} \) If we let the width be \( x \), we then have: \( 36 = (2x + 6)x \) Expanding it gives us: \( 36 = 2x^2 + 6x \) This simplifies to: \( 2x^2 + 6x - 36 = 0 \) Dividing everything by 2 gives: \( x^2 + 3x - 18 = 0 \). Now, using the quadratic formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \), where \( a = 1 \), \( b = 3 \), and \( c = -18 \), we get: \( x = \frac{{-3 \pm \sqrt{{3^2 - 4 \cdot 1 \cdot (-18)}}}}{2 \cdot 1} \) \( x = \frac{{-3 \pm \sqrt{{9 + 72}}}}{2} \) \( x = \frac{{-3 \pm \sqrt{81}}}{2} \) \( x = \frac{{-3 \pm 9}}{2} \). Thus, \( x = 3 \) (the width) or \( x = -6 \) (which we discard since it’s not feasible). Now substituting this back into the length equation: Length = \( 2(3) + 6 = 12 \) feet. So the dimensions are: Width = 3 feet Length = 12 feet Now, let's keep the fun going! In ancient architectural practices, knowing the precise calculations for area and dimensions was critical! Imagine builders of the past using sticks, stones, or clay tablets to measure areas, maximizing barn space with clever dimensions. They didn’t just wing it; they subtly combined geometry and arithmetic, principles still celebrated in today's modern engineering! Next time you walk past a barn, imagine its builders standing around, scratching their heads and measuring with rudimentary tools, hoping to one day expand their farming empire. This historical ingenuity is ever-present, even if we now have calculators and advanced software!