Tunnel Vision
How wide is this tunnel?
Age
14 to 16
Challenge level
Problem
A tunnel is cut through a hillside, with a semi-circular cross-section.
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A truck of height 6 m can just drive with its nearside wheels 2 m from the point where the curved roof meets the horizontal road surface.
How wide is the tunnel?
Student Solutions
Using a circle theorem
Triangle ABP is right-angled at P, because a triangle in a semicircle is always right-angled.
Image
Both of the two smaller right-angled triangles share an angle with triangle ABP, so the three triangles are all similar.
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The scale factor between the yellow triangle and the red triangle is 3, so the horizontal side of the red triangle is 6$\times$3=18 m. So the total width of the tunnel is 20 m.
Using Pythagoras' Theorem
OP and OB are both radii, labelled $r$, so the distance from O to the far wheel of the lorry is $r-2.$
Image
Applying Pythagoras' theorem to the right-angled triangle formed, $$\begin{align}r^2=&(r-2)^2+6^2\\
\Rightarrow r^2=&r^2-4r+40\\
\Rightarrow 0=&-4r+40\\
\Rightarrow 4r=&40\\
\Rightarrow r=&10\end{align}$$ So the width of the tunnel is $20$ m.