Question

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. a) How much of a head start did Tyrion get? b) How fast does Cersei run? 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

Hi.

a) When we look at the equations( or the graphs), Cersei's cruve starts at origin while that of Tyrion starts at +100 at y-intercept. So when Cersei started, Tyrion was +100m further. Hence, he got a 100m headstart.

This is a distance-time graph (straight line indicated uniform velocity for both), so the slope should give us their speeds.

Differentiate Tyrion's equation for displacement wrt time we get:

$\frac{d}{dt}(6t+100)=6m/s$

Time required for him to cover 100m, we find it is:

time=distance/speed

time=100/6=50/3 seconds.

So if they both started from same point, then Cersei let Tyrion either:

1. Have him start from 100m further from where she started.

OR

2. Gave him a head start of 50/3 seconds.

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b)Slope of Cersei's graph should give us her velocity.

$\frac{d}{dt}(8t)= 8m/s$

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c) Tyrion runs at 6m/s

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d) How long should the track be, for Cersei and Tyrion to end up at the finish at the same time?

Well, distance covered by both should be equal in that case.

$Cersei_{distance}=Tyrion_{distance}\\8t=6t+100\\2t=100\\t=50$

Put t=50 in any of two equations to get length of track for a tie,

8*50=400m

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e)  The significance of the solution of this linear system is that it makes our calculations easier.

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PS- Ofcourse Cersei runs faster than Tyrion ;D