A lorry and a pick-up truck travel 180 km. The pick-up truck travels 20 km/h faster than the lorry. The pick-up truck arrives 45 minutes earlier than the lorry. Find the speeds of the lorry and pick-u

Welcome back — been a while! Let's pick up where you left off. Here's the solution to the problem: Let the speed of the lorry be x km/h. The speed of the pick-up is 20 km/h faster than the lorry, so the speed of the pick-up is (x+20) km/h. The distance covered by both vehicles is 180 km. The pick-up took 45 minutes less than the lorry. Convert 45 minutes to hours: 45 min = (45)/(60) h = (3)/(4) h. a) i) Write two expressions in terms of x for the time taken by the lorry and the pick-up respectively to cover a distance of 180 km. Step 1: Write the expression for the time taken by the lorry. Time is calculated as DistanceSpeed. Time taken by lorry = (180)/(x) hours Step 2: Write the expression for the time taken by the pick-up. Time taken by pick-up = (180)/(x+20) hours The expressions are: Time for lorry: (180)/(x) hours Time for pick-up: (180)/(x+20) hours a) ii) Determine the speed of the lorry and that of the pick-up. Step 1: Set up the equation based on the time difference. The pick-up took (3)/(4) hours less than the lorry, so: Time taken by lorry - Time taken by pick-up = (3)/(4) (180)/(x) - (180)/(x+20) = (3)/(4) Step 2: Find a common denominator and combine the terms on the left side. (180(x+20) - 180x)/(x(x+20)) = (3)/(4) (180x + 3600 - 180x)/(x^2 + 20x) = (3)/(4) (3600)/(x^2 + 20x) = (3)/(4) Step 3: Cross-multiply to eliminate the denominators. 4 × 3600 = 3(x^2 + 20x) 14400 = 3x^2 + 60x Step 4: Rearrange the equation into a standard quadratic form ax^2 + bx + c = 0. 3x^2 + 60x - 14400 = 0 Step 5: Divide the entire equation by 3 to simplify. x^2 + 20x - 4800 = 0 Step 6: Solve the quadratic equation by factoring. We need two numbers that multiply to -4800 and add to 20. These numbers are 80 and -60. (x+80)(x-60) = 0 Step 7: Find the possible values for x. x+80 = 0 x = -80 x-60 = 0 x = 60 Since speed cannot be negative, we discard x = -80. Step 8: Determine the speeds of the lorry and the pick-up. Speed of the lorry (x): 60 km/h Speed of the pick-up (x+20): 60 + 20 = 80 km/h The speed of the lorry is 60 km/h and the speed of the pick-up is 80 km/h. Send me the next one 📸