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
Step 1: Define the variables and formulas. Let D be the distance, S be the speed, and T be the time. The relationship is T = (D)/(S). The distance for each leg of the journey is D = 300 km. The speed from Tema to Takoradi is S_1 = (x+10) km/h. The speed from Takoradi to Tema (return) is S_2 = (x-10) km/h. The total time for the round trip is T_total = 8 hours. Step 2: Express the time for each leg of the journey. Time taken to travel from Tema to Takoradi (T_1): T_1 = (D)/(S_1) = (300)/(x+10) Time taken to travel from Takoradi to Tema (T_2): T_2 = (D)/(S_2) = (300)/(x-10) Step 3: Set up the equation for the total time. The total time is the sum of the times for each leg: T_1 + T_2 = T_total (300)/(x+10) + (300)/(x-10) = 8 Step 4: Solve the equation for x. Multiply the entire equation by the common denominator (x+10)(x-10) to eliminate the fractions: 300(x-10) + 300(x+10) = 8(x+10)(x-10) Expand the terms: 300x - 3000 + 300x + 3000 = 8(x^2 - 10^2) 600x = 8(x^2 - 100) 600x = 8x^2 - 800 Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0: 8x^2 - 600x - 800 = 0 Divide the entire equation by 8 to simplify: x^2 - 75x - 100 = 0 Step 5: Use the quadratic formula to find the values of x. The quadratic formula is x = -b ± sqrt(b^2 - 4ac)2a. For x^2 - 75x - 100 = 0, we have a=1, b=-75, and c=-100. x = -(-75) ± sqrt((-75)^2 - 4(1)(-100))2(1) x = 75 ± sqrt(5625 + 400)2 x = 75 ± sqrt(6025)2 Step 6: Evaluate the possible values for x. x_1 = 75 + sqrt(6025)2 x_2 = 75 - sqrt(6025)2 Since sqrt(6025) ≈ 77.62, we have: x_1 ≈ (75 + 77.62)/(2) = (152.62)/(2) = 76.31 x_2 ≈ (75 - 77.62)/(2) = (-2.62)/(2) = -1.31 Speed cannot be negative. If x = -1.31, then x-10 = -1.31-10 = -11.31, which is a negative speed. Therefore, x_2 is not a valid solution. The valid value for x is x_1. The value of x is 75 + sqrt(6025)2. That's 2 down. 3 left today — send the next one.