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 — b) Step 1: Solve the equation 2^x × 8 = 64. Divide both sides by 8. 2^x = (64)/(8) 2^x = 8 Step 2: Express 8 as a power of 2. 8 = 2^3 Step 3: Substitute this back into the equation. 2^x = 2^3 Step 4: Since the bases are the same, equate the exponents. x = 3 The solution is x=3. c) i) The total time for the journey Step 1: Calculate the time for the first part of the journey. Distance = 180 km Speed = 60 km/h Time = DistanceSpeed Time_1 = 180 km60 km/h = 3 hours Step 2: Calculate the time for the second part of the journey. Distance = 150 km Speed = 50 km/h Time_2 = 150 km50 km/h = 3 hours Step 3: Calculate the total time for the journey. Total Time = Time_1 + Time_2 = 3 hours + 3 hours = 6 hours The total time for the journey is 6 hours. ii) The average speed for the entire journey Step 1: Calculate the total distance for the journey. Total Distance = 180 km + 150 km = 330 km Step 2: Calculate the average speed. Average Speed = Total DistanceTotal Time Average Speed = 330 km6 hours = 55 km/h The average speed for the entire journey is 55 km/h. a) Step 1: Find the least common multiple (LCM) of the denominators 6, 12, and 4. The LCM of 6, 12, and 4 is 12. Step 2: Convert each fraction to an equivalent fraction with a denominator of 12. (5)/(6) = (5 × 2)/(6 × 2) = (10)/(12) (7)/(12) (1)/(4) = (1 × 3)/(4 × 3) = (3)/(12) Step 3: Perform the addition and subtraction. (10)/(12) + (7)/(12) - (3)/(12) = (10 + 7 - 3)/(12) = (17 - 3)/(12) = (14)/(12) Step 4: Simplify the fraction. (14)/(12) = (14 ÷ 2)/(12 ÷ 2) = (7)/(6) The simplified expression is (7)/(6). b) Step 1: Determine the total number of ratio parts. Ratio of boys to girls is 7:8. Total ratio parts = 7 + 8 = 15. Step 2: Calculate the value of one ratio part. Total learners = 60. Value of one part = Total learnersTotal ratio parts = (60)/(15) = 4. Step 3: Calculate the number of boys. Number of boys = 7 × 4 = 28. Step 4: Calculate the number of girls. Number of girls = 8 × 4 = 32. There are 28 boys and 32 girls. c) Given: radius r = 3 m, height h = 5 m, = 3.142. i) The volume of the tank Step 1: Use the formula for the volume of a cylinder, V = r^2 h. V = 3.142 × (3 m)^2 × 5 m V = 3.142 × 9 m^2 × 5 m V = 3.142 × 45 m^3 V = 141.39 m^3 The volume of the tank is 141.39 m^3. ii) The surface area of the tank Step 1: Use the formula for the surface area of a closed cylinder, A = 2 r (r+h). A = 2 × 3.142 × 3 m × (3 m + 5 m) A = 2 × 3.142 × 3 m × 8 m A = 6.284 × 24 m^2 A = 150.816 m^2 The surface area of the tank is 150.816 m^2. Got more? Send 'em!