A rectangle’s length is 5 meters more than twice its width. If the perimeter is 58 meters, what is the area of the rectangle?
["Rectangle Length and Width: A Step-by-Step Guide to Solving the Perimeter and Area Problem", "Understanding the relationship between the length and width of a rectangle and how the perimeter affects its area is a key concept in geometry. In this article, we’ll solve a classic problem: determining the area of a rectangle when one side’s length depends on the other, and the total perimeter is given.", "Problem Statement:", "A rectangle’s length is 5 meters more than twice its width. If the perimeter is 58 meters, what is the area of the rectangle?", "---", "### Step 1: Define Variables", "First, assign variables to represent the unknown dimensions:", "- Let ( w ) = width of the rectangle (in meters)\n- Then, length ( l = 2w + 5 ) (since length is 5 more than twice the width)", "---", "### Step 2: Use the Perimeter Formula", "The perimeter ( P ) of a rectangle is given by:", "[\nP = 2 \ imes (\ ext{length} + \ ext{width}) = 2(l + w)\n]", "We know the perimeter is 58 meters, so:", "[\n2(l + w) = 58\n]", "Substitute ( l = 2w + 5 ):", "[\n2((2w + 5) + w) = 58\n]", "---", "### Step 3: Solve for Width", "Simplify the equation:", "[\n2(3w + 5) = 58\n]\n[\n6w + 10 = 58\n]\n[\n6w = 58 - 10 = 48\n]\n[\nw = \frac{48}{6} = 8 \ ext{ meters}\n]", "---", "### Step 4: Find the Length", "Now use ( l = 2w + 5 ):", "[\nl = 2(8) + 5 = 16 + 5 = 21 \ ext{ meters}\n]", "---", "### Step 5: Calculate the Area", "The area ( A ) of a rectangle is given by:", "[\nA = \ ext{length} \ imes \ ext{width} = 21 \ imes 8 = 168 \ ext{ square meters}\n]", "---", "### Summary", "- Width = 8 meters\n- Length = 21 meters\n- Perimeter = 58 meters (checked via ( 2(8 + 21) = 58 ))\n- Area = 168 m²", "---", "### Final Answer", "The area of the rectangle is 168 square meters.", "This step-by-step approach shows how algebraic reasoning connects linear relationships and geometric measurements, making it easier to solve real-world problems involving shapes and their properties. Whether you're a student learning geometry or a builder calculating materials, understanding perimeter and area relationships is essential."]
