Two cyclists, Aiko and Ben, are riding along the same straight track in the same direction.
Aiko starts 2000 m ahead of Ben. Aiko rides at a constant speed of 250 metres/minute and Ben rides at a constant speed of 500 metres/minute.
Let \(d\) = distance from the starting point in metres, and
\(t\) = time in minutes.
- The equation \(d=2000+250t\) models Aiko's distance from the starting point.
- Write an equation to model Ben's distance. (1 mark)
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- Use the equations from (a) to graph Aiko and Ben's journeys on the grid below. (2 marks)
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- After how many minutes does Ben catch Aiko? (1 mark)
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- How far has each cyclist travelled from their own starting point when they meet? (2 marks)
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a. \(d=500t\)
b.
c. \(8\ \text{minutes}\)
d. \(\text{Aiko travelled 2000 m, Ben travelled 4000 m.}\)
a. \(d=500t\)
b. \(\text{Table of values:}\)
\(\begin{array}{|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \text{Aiko} & 2000 & 2500 & 3000 & 3500 & 4000 & 4500 \\ \hline \text{Ben} & 0 & 1000 & 2000 & 3000 & 4000 & 5000 \\ \hline \end{array}\)
c. \(\text{From the graph, the lines intersect at }\ t=8.\)
\(\therefore\ \text{Ben catches Aiko after 8 minutes}\)
d. \(\text{When}\ \ t=8, d=4000:\)
\(\text{Aiko started 2000 m ahead, so the distance travelled}\)
\(=4000-2000=2000\ \text{m}\)
\(\text{Ben started at the origin, so the distance travelled =4000 m}\)