Question

Carly is a meteorologist. She is using a device called a sextant to determine the height of a weather balloon. She is holding it up to her eye, while kneeling on the ground. The sextant is exactly I meter above the ground. When she views the weather balloon through her sextant, she measures a 44° angle of elevation from the horizontal. Then, she receives a radio signal from the balloon which tells her that the balloon is 1400 meters directly from the sextant. How high is the balloon? How far is it to a position directly beneath the balloon?

Answer 1

Let's draw a picture of our problem:

where h is the height of the ballon. Since we have a right triangle, we can use a trigonometric function to relate the height, the distance from the observer to the ballon and the given angle. This function is the sine function, that is

$\sin (44)=\frac{h}{1400}$

so, by moving 1400 to the left hand side, we get

$1400\cdot\sin (44)=h$

then, h is given by

$\begin{gathered} h=1400\cdot sin(44)\text{ m} \\ h=972.52\text{m} \end{gathered}$

Now, lets find the position directly beneath the balloon. This is given by d in our picture. Then, we must relate d with the given angle and 1400 m. This function is the cosine function of 44 degrees. that is,

$\cos (44)=\frac{d}{1400}$

hence, by moving 1400 to the left hand side, we have

$1400\cdot\cos (44)=d$

so, d is given by

$\begin{gathered} d=1400\cdot\cos (44)\text{ m} \\ d=1007.08\text{ m} \end{gathered}$

Therefore, the answers are:

How high is the balloon? 972.52 meters

How far is it to a position directly beneath the balloon?​ 1007.08 meters