A train travels away from a station. Its distance, \(d\) km, from the station after \(t\) minutes is given by \(d = 1.5t\) for \(0 \leq t \leq 20\). The tab...
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A train travels away from a station. Its distance, \(d\) km, from the station after \(t\) minutes is given by \(d = 1.5t\) for \(0 \leq t \leq 20\). The table below shows some values.
| \(t\) | 0 | 5 | 10 | 15 | 20 |
|---|---|---|---|---|---|
| \(d\) | 0 | 15 | 30 |
- Fill in the missing entries in the table. (2)
- Draw the graph of \(d = 1.5t\) on the grid below for \(0 \leq t \leq 20\). (2)
- Read from your line the distance travelled after 12 minutes. (1)
Answer Details
The relationship \(d = 1.5t\) is directly proportional: the graph is a straight line through the origin with gradient \(1.5\). To complete the table, substitute each given value of \(t\); to draw the graph, plot the table's points and join them with a straight line since the relationship is linear.
- Substituting into \(d = 1.5t\): \(d(5) = 1.5 \times 5 = 7.5\) and \(d(15) = 1.5 \times 15 = 22.5\). The completed table is:
[2] (both missing values correct)\(t\) 0 5 10 15 20 \(d\) 0 7.5 15 22.5 30 - Plotting these five points and joining them gives a straight line from \((0,0)\) to \((20,30)\), since the train travels at a constant \(1.5\) km per minute: [2] (straight line correctly drawn through the plotted points)
- Reading up from \(t=12\) on the horizontal axis to the line, then across to the vertical axis (the dashed guide lines), gives \(d = 18\) km. This agrees with calculating directly: \(d = 1.5 \times 12 = 18\) km. [1]
A directly proportional graph always passes through the origin; once you have one other point, or the gradient, you can extend the line accurately without needing every table value plotted first.
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