Question

show work, 3 decimal places, calculator ready form

1. as a hot a air balloon rises vertically, its angle of elevation from your perspective on level ground 10 kilometres away from its takeoff position changes from 19° 20 to 31° 0. approximately how far does the balloon rise during this period? what should the riders in said balloon take with them based on your conclusion? draw diagram for full credit.

Explanation:

Step1: Convert angles to decimal degrees

First, convert \(19^\circ 20'\) and \(31^\circ 0'\) to decimal degrees. Recall that \(1^\circ = 60'\), so \(20'=\frac{20}{60}\approx0.333^\circ\), so \(19^\circ 20'\approx19.333^\circ\), and \(31^\circ 0' = 31.0^\circ\).

Step2: Use tangent function for height

Let \(d = 10\) km (distance from observer to take - off point), \(h_1\) be the initial height, and \(h_2\) be the final height. The tangent of the angle of elevation \(\theta\) is given by \(\tan\theta=\frac{h}{d}\), so \(h = d\tan\theta\).

For the initial angle \(\theta_1 = 19.333^\circ\), \(h_1=d\tan\theta_1=10\times\tan(19.333^\circ)\). Calculate \(\tan(19.333^\circ)\approx\tan(19 + \frac{20}{60})\approx0.350\) (using calculator: \(\tan(19.333^\circ)\approx\tan(19.333)\approx0.3502\)). So \(h_1 = 10\times0.3502 = 3.502\) km.

For the final angle \(\theta_2=31.0^\circ\), \(\tan(31.0^\circ)\approx0.6009\). So \(h_2 = 10\times0.6009 = 6.009\) km.

Step3: Calculate the rise in height

The rise in height \(\Delta h=h_2 - h_1\). Substitute the values: \(\Delta h=6.009 - 3.502=2.507\) km (rounded to 3 decimal places).

(For the second part about what riders should take: As the balloon rises to a higher altitude, the temperature drops and the air pressure decreases. So riders should take warm clothing (due to lower temperatures at higher altitudes) and maybe oxygen supplies (if the altitude is high enough where air pressure is too low for normal breathing, although for a hot - air balloon, the typical altitudes may not require oxygen but warm clothing is a must as temperature decreases with altitude). But the main calculation part is about the height rise.)

Answer:

The balloon rises approximately \(\boldsymbol{2.507}\) kilometers. (For the second part, riders should take warm clothing (and potentially other items like sunscreen, etc. depending on altitude, but warm clothing is a key item due to temperature drop with altitude).)