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Let \(A\) and \(B\) be endpoints of the single lane.
Let \(O\) be the midpoint of \(\overline{AB}\).
The single lane is \(12\) ft wide and equidistant from sides of tunnel, which means its present right in middle of the tunnel.
Now, we know that length of \(\overline{AB} = 12\)ft, which means \(\overline{AO} = \overline{OB} = \frac{\overline{12}}{2} = 6\)ft.
Since there a mention of
height of vehicles, draw a perpendicular from either of point \(A\) or \(B\)(lets take \(B\)) till it meets the curved circumference of semicircle at point \(C\).
Why from point \(B\)? Because that is where the lane ends, vehicles cannot travel past that.
This basically forms a right-angled triangle \(OBC\) with \(OC\) as hypotenuse as well as radius of the semicircle = \(\frac{20}{2} = 10\)ft.
So max height possible = \(BC - \frac{1}{2} = \sqrt{OC^2 - OB^2} - \frac{1}{2} = \sqrt{10^2 - 6^2} - \frac{1}{2} = \sqrt{64} - \frac{1}{2} = 8 - \frac{1}{2} = 7\frac{1}{2}\)ft
\((-\frac{1}{2})\) because the question states a condition -
vehicles must clear the top of the tunnel by at least 1/2 footHence, Answer is
B