Question

9. Cersei and her brother Tyrion decide to race home. Cersei is a faster runner than Tyrion, so she gives him a head start. Their distance- time graphs are shown. de Cersei and Tyrion's Race Home 900 800 E700 600 500 4000 300 200 100 0 Distance (m) Cersei d=8t Tyrion d=6t+100 20 40 60 80 100 Time (s) a) How much of a head start did Tyrion get? b) How fast does Cersei run? 300 50 20 c) How fast does Tyrion run? d) For what length of race will each runner win? For what length of race will they tie? e) Explain the significance of the solution of this linear system.​

Answer 1

Explanation:

a) To determine the head start Tyrion got, we need to find the value of the y-intercept for his distance-time graph. In the equation for Tyrion's graph, d = 6t + 100, the constant term 100 represents the y-intercept. Therefore, Tyrion got a head start of 100 meters.

b) To find Cersei's running speed, we can look at the equation for her distance-time graph, which is given as d = 8t. In this equation, the coefficient of t (8) represents Cersei's running speed. Therefore, Cersei runs at a speed of 8 meters per second.

c) Tyrion's running speed can be determined from the equation for his distance-time graph, d = 6t + 100. Here, the coefficient of t (6) represents Tyrion's running speed. Therefore, Tyrion runs at a speed of 6 meters per second.

d) To determine the length of the race that each runner wins, we need to compare their distance-time equations. Cersei's equation is d = 8t, while Tyrion's equation is d = 6t + 100.

For Cersei to win, her distance must be greater than Tyrion's distance. Setting their equations equal to each other, we get 8t = 6t + 100. Solving for t, we find t = 50. This means that if the race lasts for 50 seconds or less, Cersei will win.

For Tyrion to win, his distance must be greater than Cersei's distance. Setting their equations equal to each other, we get 6t + 100 = 8t. Solving for t, we find t = 50. This means that if the race lasts for more than 50 seconds, Tyrion will win.

To determine the length of the race where they tie, we set their equations equal to each other and solve for t. Setting 8t = 6t + 100, we find t = 25. This means that if the race lasts for exactly 25 seconds, Cersei and Tyrion will tie.

e) The solution to this linear system represents the points in time where Cersei and Tyrion's distances are equal, where one runner is ahead, and where they tie. It helps us understand the relationship between their running speeds and head start. By analyzing the solution, we can determine the conditions under which each runner wins or ties. Additionally, this linear system provides a mathematical representation of their race, allowing us to make predictions and analyze the race's outcomes.

Hope this helps.