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.
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You're on a roll — here are the solutions to all the questions on this paper: 6. a) The travel graph below describes the journey of a cyclist from Town X to Town Y. Study the graph and use it to answer the questions that follow. i. What is the average speed for the return journey from Town Y to Town X? Step 1: Identify the start and end points of the return journey. The return journey starts at Town Y (60 km from Town X) at 3 pm and ends at Town X (0 km) at 4 pm. Step 2: Calculate the distance covered. Distance = 60 km - 0 km = 60 km Step 3: Calculate the time taken. Time = 4 pm - 3 pm = 1 hour Step 4: Calculate the average speed. Average Speed = DistanceTime = 60 km1 hour = 60 km/h The average speed for the return journey is 60 km/h. ii. State the period within which he travelled to Town Y after his first rest? Step 1: Identify the first rest period. The graph shows a horizontal line segment from 1 pm to approximately 1:30 pm at a distance of 35 km, indicating a rest. Step 2: Identify the travel segment to Town Y after this rest. After 1:30 pm, the cyclist resumes travel. Town Y is at 60 km. The cyclist reaches 60 km at 2:30 pm. The period of travel to Town Y after his first rest is from 1:30 pm to 2:30 pm. The period is 1:30 pm to 2:30 pm. iii. How many minutes did the cyclist spend at Town Y? Step 1: Identify the arrival time at Town Y. The cyclist arrives at Town Y (60 km) at 2:30 pm. Step 2: Identify the departure time from Town Y. The cyclist starts the return journey from Town Y (60 km) at 3 pm. Step 3: Calculate the time spent at Town Y. Time spent = 3 pm - 2:30 pm = 30 minutes The cyclist spent 30 minutes at Town Y. iv. What distance did the cyclist cover between 1pm and 2pm? Step 1: Determine the distance at 1 pm. At 1 pm, the distance from Town X is 35 km. Step 2: Determine the distance at 2 pm. At 2 pm, the distance from Town X is 55 km. Step 3: Calculate the distance covered between 1 pm and 2 pm. Distance covered = 55 km - 35 km = 20 km The cyclist covered 20 km between 1 pm and 2 pm. 6. b) Copy and complete the table below | Value | Number of significant figures | |---|---| | 142.431 | 6 | | 0.000562 | 3 | | 5.32 x 10^5 | 3 | | 64800002 | 8 | | 1.0072 x 10^2 | 5 | Explanation: 142.431: All non-zero digits are significant. There are 6 non-zero digits. 0.000562: Leading zeros (0.000) are not significant. The non-zero digits 5, 6, 2 are significant. 5.32 x 10^5: In scientific notation, all digits in the mantissa (5.32) are significant. 64800002: All non-zero digits (6, 4, 8, 2) are significant. Zeros between non-zero digits are also significant. 1.0072 x 10^2: In scientific notation, all digits in the mantissa (1.0072) are significant, including the zeros between non-zero digits. 6. c) Evaluate: 1 - (2)/(3) - ((3)/(4) + 2 - (5)/(8)) Step 1: Evaluate the expression inside the parenthesis first. (3)/(4) + 2 - (5)/(8) Find a common denominator for 4, 1 (from 2 = (2)/(1)), and 8. The least common multiple is 8. (3 × 2)/(4 × 2) + (2 × 8)/(1 × 8) - (5)/(8) (6)/(8) + (16)/(8) - (5)/(8) (6 + 16 - 5)/(8) = (22 - 5)/(8) = (17)/(8) Step 2: Substitute this result back into the main expression. 1 - (2)/(3) - (17)/(8) Find a common denominator for 1, 3, and 8. The least common multiple is 24. (1 × 24)/(24) - (2 × 8)/(3 × 8) - (17 × 3)/(8 × 3) (24)/(24) - (16)/(24) - (51)/(24) Step 3: Combine the fractions. (24 - 16 - 51)/(24) (8 - 51)/(24) (-43)/(24) The evaluated expression is -(43)/(24). Got more? Send 'em!