Traffic tunnel
Will these vehicles fit through this tunnel?
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.
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
How high is the tunnel at $x=1, x=1.5, x=2$?
$x=1\Rightarrow y=\frac49(9-1)=\frac{32}9=3\frac59=3.\dot5$
$x=1.5\Rightarrow y=\frac49(9-1.5^2)= \frac49\left(\frac{36}4-\frac94\right) = 4-1=3$
$x=2\Rightarrow y=\frac49(9-2^2)=\frac49\times5=2\frac29=2.\dot2$
All of the vehicles are too tall (or too wide) for the tunnel:
$3.9\gt3.\dot5$
$3.4\gt3$
$2.7\gt2.\dot2$