This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
i. Write an expression for the length of the garden. Step 1: Define the given width. The width of the garden is represented by w meters. Step 2: Formulate the length based on the problem description. The length is 5 meters longer than its width. Length = width + 5 L = w + 5 The expression for the length is w + 5 meters. ii. Write an expression for the perimeter of the garden. Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P of a rectangle is given by P = 2(length + width). Step 2: Substitute the expressions for length and width into the formula. We have L = w + 5 and width = w. P = 2((w + 5) + w) Step 3: Simplify the expression. P = 2(2w + 5) P = 4w + 10 The expression for the perimeter is 4w + 10 meters. iii. Write an expression for the area of the garden. Step 1: Recall the formula for the area of a rectangle. The area A of a rectangle is given by A = length × width. Step 2: Substitute the expressions for length and width into the formula. We have L = w + 5 and width = w. A = (w + 5)w Step 3: Simplify the expression. A = w^2 + 5w The expression for the area is w^2 + 5w square meters. c. If the width of the garden is 8 meters, calculate the perimeter and area. Step 1: Substitute w = 8 meters into the perimeter expression from part (ii). P = 4w + 10 P = 4(8) + 10 P = 32 + 10 P = 42 meters The perimeter is 42 meters. Step 2: Substitute w = 8 meters into the area expression from part (iii). A = w^2 + 5w A = (8)^2 + 5(8) A = 64 + 40 A = 104 square meters The area is 104 square meters. What's next?