Question

A​ hot-air balloon is rising vertically. The angle of elevation from a point on level ground 123 feet from the balloon to a point directly under the passenger compartment changes from 19.6 degrees to 29.4 degrees. How​ far, to the nearest tenth of a​ foot, does the balloon rise during this​ period?

Answer 1

Answer:

25.5 feet

Explanation:

we are given the following:

initial angle of elevation = 19.6 degrees

final angle of elevation = 29.4 degrees

distance from the point of elevation = 123 feet

find the increase in height (rise) of the balloon ?

we should first take not that the hot air balloon,  the point of elevation on the ground and the point directly under the passenger form a right angel triangle with:

• The distance between the hot air balloon and the point of elevation on the ground forming the hypotenuse
• The distance between the point of elevation and the point directly under the passenger forming the adjacent side
• The distance between the point directly under the passenger and the hot air balloon forming the opposite side
• The angle of elevation is the angle formed between the adjacent side and the hypotenuse ( the hot air balloon, the point of elevation and the point directly under the passenger)
• The increase in height would be the height at the final angle elevation minus the height at the initial angle of elevation.

This heights can be gotten from applying the phytaghoras theorem where

tan (angle of elevation) =\frac{height}{adjacent side}

height = adjacent side × tan (angle if elevation)

(adjacent side is the distance from the point of elevation = 123 feet while the angle of elevation = 19.6 degrees and 29.4 degrees )

• At the final angle of elevation

             height = 123 x tan 29.4 = 69.3 feet

• At initial angle of elevation

             height = 123 x tan 19.6 = 43.8 feet

increase in height =69.3 - 43.8 = 25.5 feet