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
You're on a roll — a) Calculate the value of x. Step 1: Determine the dimensions of the entire room. The dancing space has dimensions (x-3) m by (x+3) m. The carpeted margin is (2)/(5)x wide all around. The total length of the room is the length of the dancing space plus twice the margin width: L_total = (x+3) + 2((2)/(5)x) = x+3 + (4)/(5)x = (5x+15+4x)/(5) = (9x+15)/(5) The total width of the room is the width of the dancing space plus twice the margin width: W_total = (x-3) + 2((2)/(5)x) = x-3 + (4)/(5)x = (5x-15+4x)/(5) = (9x-15)/(5) Step 2: Use the total area to form an equation for x. The total area of the entire room is 315 m^2. A_total = L_total × W_total 315 = ((9x+15)/(5)) ((9x-15)/(5)) 315 = ((9x)^2 - (15)^2)/(25) 315 = (81x^2 - 225)/(25) Step 3: Solve the equation for x. Multiply both sides by 25: 315 × 25 = 81x^2 - 225 7875 = 81x^2 - 225 Add 225 to both sides: 7875 + 225 = 81x^2 8100 = 81x^2 Divide by 81: x^2 = (8100)/(81) x^2 = 100 Take the square root: x = ±sqrt(100) Since x represents a dimension, it must be positive. x = 10 The value of x is 10. b) Calculate the area of the carpeted margin. Step 1: Calculate the area of the dancing space. Using x=10: Length of dancing space L_dancing = x+3 = 10+3 = 13 m Width of dancing space W_dancing = x-3 = 10-3 = 7 m Area of dancing space A_dancing = L_dancing × W_dancing = 13 m × 7 m = 91 m^2. Step 2: Calculate the area of the carpeted margin. The area of the carpeted margin is the total area minus the area of the dancing space. A_margin = A_total - A_dancing A_margin = 315 m^2 - 91 m^2 A_margin = 224 m^2 The area of the carpeted margin is 224 m^2. c) If the carpet cost sh. 750 per m^2, calculate the total cost of the sealed margin. The "sealed margin" refers to the carpeted margin. Cost = Area of carpeted margin × Cost per m^2 Cost = 224 m^2 × Sh. 750/m^2 Cost = Sh. 168000 The total cost of the sealed margin is Sh. 168,000. Got more? Send 'em!