Traffic tunnel
Will these vehicles fit through this tunnel?
Age
14 to 16
Challenge level
Problem
A traffic tunnel is being planned. The diagram shows the cross-section of the tunnel. The shape of the tunnel is the same as the graph of $y=\frac49(9-x^2)$.
All distances are in metres.
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Which of these vehicles will fit through the tunnel?
a) truck, 2.9 m high, 4 m wide
b) coach, 3.4 m high, 3 m wide
c) lorry, 3.9 m high, 2 m wide
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: None of the vehicles will fit through the tunnel.
a) truck b) coach c) lorry
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How high is the tunnel at $x=1, x=1.5, x=2$?
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$x=1.5\Rightarrow y=\frac49(9-1.5^2)= \frac49\left(\frac{36}4-\frac94\right) = 4-1=3$
$x=2\Rightarrow y=\frac49(9-2^2)=\frac49\times5=2\frac29=2.\dot2$
All of the vehicles are too tall (or too wide) for the tunnel:
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$3.9\gt3.\dot5$
$3.4\gt3$
$2.7\gt2.\dot2$