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
Here are the solutions to the questions based on the provided graph and equation. a) i. Step 1: Identify the starting point of the journey (distance from home is 0 km). The graph starts at (8 a.m., 0 km). The cyclist left home at 8 a.m. ii. Step 1: Identify the point where the cyclist returns home (distance from home is 0 km). The graph ends at (5 p.m., 0 km). The cyclist returned home at 5 p.m. iii. Step 1: Locate 10 a.m. on the time axis and find the corresponding distance. At 10 a.m., the distance from home was 20 km. iv. Step 1: Locate 11 a.m. on the time axis and find the corresponding distance. At 11 a.m., the distance from home was 20 km. v. Step 1: Locate 2 p.m. on the time axis and find the corresponding distance. At 2 p.m., the distance from home was 60 km. vi. Step 1: Locate 3 p.m. on the time axis and find the corresponding distance. At 3 p.m., the distance from home was 60 km. vii. Step 1: Identify horizontal segments on the graph, which indicate no change in distance over time. The cyclist took a rest from 10 a.m. to 11 a.m. and from 2 p.m. to 3 p.m. viii. Step 1: Locate noon (12 p.m.) on the time axis and find the corresponding distance. At noon, the distance from home was 40 km. ix. Step 1: Locate 5 p.m. on the time axis and find the corresponding distance. At 5 p.m., the distance from home was 0 km. x. Step 1: Calculate the speed (gradient) for each moving segment. Speed is the magnitude of the gradient of the distance-time graph. Segment 1 (8 a.m. to 10 a.m.): Speed_1 = 20 km - 0 km10 a.m. - 8 a.m. = 20 km2 h = 10 km/h Segment 2 (11 a.m. to 2 p.m.): Speed_2 = 60 km - 20 km2 p.m. - 11 a.m. = 40 km3 h ≈ 13.33 km/h Segment 3 (3 p.m. to 5 p.m.): Speed_3 = | 0 km - 60 km5 p.m. - 3 p.m. | = | -60 km2 h | = 30 km/h Step 2: Compare the speeds to find the fastest. The speeds are 10 km/h, 13.33 km/h, and 30 km/h. The fastest speed is 30 km/h. The cyclist was travelling most quickly from 3 p.m. to 5 p.m. b) Step 1: Rearrange the equation 6x - 2y - 4 = 0 into the slope-intercept form y = mx + c. 6x - 2y - 4 = 0 -2y = -6x + 4 Step 2: Divide by -2. y = (-6x)/(-2) + (4)/(-2) y = 3x - 2 Step 3: Identify the gradient (m) and y-intercept (c). The gradient is m = 3. The y-intercept is c = -2. The gradient is 3 and the y-intercept is -2. Send me the next one 📸