Ship crossing a radar coverage circle
Ship crossing a radar coverage circle
IP 3
10 marks
On a chart with units in kilometres, a radar at \(R(2,1)\) detects ships within \(5\) km. A ship sails north-east along the line \(y=x+3\) at a steady \(12\) km/h. Model the ship as a point.
- Find, in exact form, the coordinates of the points where the ship enters and leaves the radar’s coverage.[3]
- Find the distance the ship travels inside the coverage and the time, to the nearest minute, for which it is detected.[3]
- A second ship sails along the line \(y=x+c\). Find the range of values of \(c\) for which it is never detected by the radar.[4]
Solution:
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Answer:(a) \(\left(-\dfrac{\sqrt{34}}{2},\ 3-\dfrac{\sqrt{34}}{2}\right)\) and \(\left(\dfrac{\sqrt{34}}{2},\ 3+\dfrac{\sqrt{34}}{2}\right)\) (b) \(2\sqrt{17}\approx 8.25\) km, about \(41\) min (c) \(c<-1-5\sqrt{2}\) or \(c>-1+5\sqrt{2}\)
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