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
a) Calculate the radius of the lawn Step 1: Define the radii and areas. Let r be the radius of the lawn. The area of the lawn is A_lawn = r^2. The path has a uniform width of 7 m, so the outer radius (lawn + path) is R = r + 7. The total area (lawn + path) is A_total = (r+7)^2. The area of the path is A_path = A_total - A_lawn = (r+7)^2 - r^2. Step 2: Set up the equation based on the given information. We are given that the area of the path is 21\% of the area of the lawn. A_path = 0.21 × A_lawn (r+7)^2 - r^2 = 0.21 r^2 Step 3: Simplify the equation by dividing by and expanding. (r+7)^2 - r^2 = 0.21 r^2 r^2 + 14r + 49 - r^2 = 0.21 r^2 14r + 49 = 0.21 r^2 Step 4: Rearrange into a quadratic equation and solve for r. 0.21 r^2 - 14r - 49 = 0 Multiply by 100 to clear decimals: 21 r^2 - 1400r - 4900 = 0 Divide by 7: 3 r^2 - 200r - 700 = 0 Using the quadratic formula r = -b ± sqrt(b^2 - 4ac)2a: r = -(-200) ± sqrt((-200)^2 - 4(3)(-700))2(3) r = 200 ± sqrt(40000 + 8400)6 r = 200 ± sqrt(48400)6 r = (200 ± 220)/(6) We get two possible values for r: r_1 = (200 + 220)/(6) = (420)/(6) = 70 r_2 = (200 - 220)/(6) = (-20)/(6) = -(10)/(3) Since the radius cannot be negative, we take the positive value. The radius of the lawn is 70 m. b) Calculate the total number of posts required for the fencing Step 1: Calculate the circumference of the inner and outer sides. For this problem, we will use = (22)/(7) for exact integer results for the number of posts. Inner radius r = 70 m. Inner circumference C_inner = 2 r = 2 × (22)/(7) × 70 = 2 × 22 × 10 = 440 m. Outer radius R = r + 7 = 70 + 7 = 77 m. Outer circumference C_outer = 2 R = 2 × (22)/(7) × 77 = 2 × 22 × 11 = 484 m. Step 2: Calculate the number of posts for each side. Posts on the inner side are at intervals of 10 m: N_inner = C_inner10 = (440)/(10) = 44 posts Posts on the outer side are at intervals of 11 m: N_outer = C_outer11 = (484)/(11) = 44 posts Step 3: Calculate the total number of posts. Total posts = N_inner + N_outer = 44 + 44 = 88 posts. The total number of posts required is 88 posts. c) Calculate the total cost of the posts Step 1: Use the total number of posts and the cost per post. Total number of posts = 88. Cost per post = sh 105. Step 2: Calculate the total cost. Total cost = Total number of posts × Cost per post Total cost = 88 × 105 = 9240 sh. The total cost of the posts is sh 9240. That's 2 down. 3 left today — send the next one.