Question

a cyclist is riding on a mountain trail. her current elevation is 1.6 km, and she is riding at a constant speed of 27 km/hr. she is riding downhill, and her angle of descent is 2° from the horizontal. see the figure below. (the figure is not drawn to scale.) find her new elevation, h, after 0.20 hours. carry your intermediate computations to at least four decimal places, and round your answer to the nearest tenth of a kilometer.

Explanation:

Step1: Calculate distance ridden

The cyclist's speed is 27 km/hr, and time is 0.20 hours. Using \( \text{distance} = \text{speed} \times \text{time} \), we get \( d = 27 \times 0.20 = 5.4 \) km.

Step2: Find vertical drop

In the right triangle, the angle of descent is \( 2^\circ \), and the hypotenuse (distance ridden) is 5.4 km. The vertical drop \( x \) is found using \( \sin(2^\circ) = \frac{x}{5.4} \), so \( x = 5.4 \times \sin(2^\circ) \). Calculating \( \sin(2^\circ) \approx 0.0348995 \), we get \( x \approx 5.4 \times 0.0348995 \approx 0.1884573 \) km.

Step3: Calculate new elevation

Initial elevation is 1.6 km. Subtract the vertical drop: \( h = 1.6 - 0.1884573 \approx 1.4115427 \) km. Rounding to the nearest tenth, \( h \approx 1.4 \) km.

Answer:

1.4