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.
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You're on a roll — 18. a) Find the width of the path. Step 1: Let the width of the path be x meters. The dimensions of the swimming pool are 10 m long and 8 m wide. The dimensions of the pool including the path will be: Length = (10 + 2x) m Width = (8 + 2x) m Step 2: Calculate the total area of the water surface and the path. The total area is given as 168 m^2. (10 + 2x)(8 + 2x) = 168 Step 3: Expand and simplify the equation. 80 + 20x + 16x + 4x^2 = 168 4x^2 + 36x + 80 = 168 4x^2 + 36x + 80 - 168 = 0 4x^2 + 36x - 88 = 0 Step 4: Divide the entire equation by 4 to simplify. x^2 + 9x - 22 = 0 Step 5: Solve the quadratic equation for x. We can factorize the quadratic equation. We need two numbers that multiply to -22 and add up to 9. These numbers are 11 and -2. (x + 11)(x - 2) = 0 This gives two possible solutions for x: x + 11 = 0 x = -11 x - 2 = 0 x = 2 Step 6: Choose the valid solution. Since the width of a path cannot be negative, we take the positive value. x = 2 The width of the path is 2 m. b) Calculate i) The number of the smaller slabs used. Step 1: Determine the area of the path. Area of the pool = 10 m × 8 m = 80 m^2. Total area of pool and path = 168 m^2. Area of the path = Total area - Area of pool Area of path = 168 m^2 - 80 m^2 = 88 m^2 Step 2: Calculate the area covered by the corner slabs. The side of each corner slab is equal to the width of the path, which is 2 m. Area of one corner slab = 2 m × 2 m = 4 m^2. There are 4 corners, so 4 corner slabs. Area covered by corner slabs = 4 × 4 m^2 = 16 m^2. Step 3: Calculate the area covered by the smaller slabs. Area covered by smaller slabs = Area of path - Area covered by corner slabs Area covered by smaller slabs = 88 m^2 - 16 m^2 = 72 m^2 Step 4: Calculate the number of smaller slabs. The side of each smaller slab is 50 cm, which is 0.5 m. Area of one smaller slab = 0.5 m × 0.5 m = 0.25 m^2. Number of smaller slabs = Area covered by smaller slabsArea of one smaller slab Number of smaller slabs = 72 m^20.25 m^2 = 288 The number of smaller slabs used is 288. ii) The total cost of the slabs used to cover the whole path. Step 1: Calculate the cost of the corner slabs. Number of corner slabs = 4. Cost of each corner slab = sh 600. Cost of corner slabs = 4 × 600 = sh 2400. Step 2: Calculate the cost of the smaller slabs. Number of smaller slabs = 288 (from part b(i)). Cost of each smaller slab = sh 50. Cost of smaller slabs = 288 × 50 = sh 14400. Step 3: Calculate the total cost. Total cost = Cost of corner slabs + Cost of smaller slabs Total cost = sh 2400 + sh 14400 = sh 16800 The total cost of the slabs used to cover the whole path is sh 16800. Got more? Send 'em!