Math  /  Calculus

Question53. (II) Uphill escape ramps are sometimes provided to the side of steep downhill highways for trucks with overheated brakes. For a simple 1111^{\circ} upward ramp, what length would be needed to stop a runaway truck traveling 140 km/h140 \mathrm{~km} / \mathrm{h} ? Note the large size of your calculated length. (If sand is used for the bed of the ramp, its length can be reduced by a factor of about 2 .)

Studdy Solution
Calculate the length of the ramp using trigonometry. The height h h and the length L L of the ramp are related by the sine of the angle:
sin(11)=hL \sin(11^\circ) = \frac{h}{L}
Solve for L L :
L=hsin(11) L = \frac{h}{\sin(11^\circ)}
Substitute h=77.08m h = 77.08 \, \text{m} :
L=77.08sin(11) L = \frac{77.08}{\sin(11^\circ)}
77.080.1908 \approx \frac{77.08}{0.1908}
403.98m \approx 403.98 \, \text{m}
The length of the ramp needed is approximately:
404m \boxed{404 \, \text{m}}

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