Question

a cyclist is riding on a mountain trail. her current elevation is 1.5 km, and she is riding at a constant speed of 22 km/hr. she is riding downhill, and her angle of descent is 4° from the horizontal. see the figure below. (the figure is not drawn to scale.) find her new elevation, h, after 0.25 hours. carry your intermediate computations to at least four decimal places, and round your answer to the nearest tenth of a kilometer. h = □ kilometers

Explanation:

Step1: Calculate the distance traveled

The formula for distance \(d = vt\), where \(v = 22\) km/hr and \(t=0.25\) hours.
\(d=22\times0.25 = 5.5\) km.

Step2: Find the change in elevation

We use the sine function. If the angle of descent \(\theta = 4^{\circ}\) and the hypotenuse (distance traveled) \(c = 5.5\) km. Let the change in elevation be \(\Delta h\). Then \(\sin\theta=\frac{\Delta h}{c}\).
\(\Delta h = c\times\sin\theta\). Substitute \(c = 5.5\) and \(\theta = 4^{\circ}\) (so \(\sin(4^{\circ})\approx0.0698\)).
\(\Delta h=5.5\times0.0698 = 0.3839\) km.

Step3: Calculate the new elevation \(h\)

The initial elevation \(H = 1.5\) km. Then \(h=H-\Delta h\).
\(h = 1.5- 0.3839=1.1161\approx1.1\) km.

Answer:

\(1.1\) kilometers