Question

three people are watching a hot air balloon travel over their town. at a certain point in time, one person stands directly below the balloon, and the others look at it at certain angles. in the following image, a,b, and c are people, and d is the balloon. person c is 384m directly below the balloon, person b is 200m away from person c, and the angle between person a, the balloon, and person b is 33 degrees. how far is person a from the hot air balloon

Answer 1

Answer:

Distance between balloon and a is = 383.67 m

Step-by-step explanation:

The given situation can be represented as the given diagram as attached in the answer area.

cd = 384 m

cb = 200 m

$\angle adb = 33^\circ$

To find:

Distance between balloon and a i.e. side ad = ?

Solution:

First of all, let us consider the right angled $\triangle bcd$.

We know the trigonometric identity that:

$tan\theta = \dfrac{Perpendicular}{Base}$

$tan\angle cbd =\dfrac{cd}{cb}\\\Rightarrowtan\angle cbd =\dfrac{384}{200}\\\Rightarrowtan\angle cbd =1.92\\\Rightarrow \angle cbd = tan^{-1}(1.92) = 62.49^\circ$

Now, using the external angle property for the external $\angle cbd$ for the $\triangle abd$:

(External angle is equal to the sum of two opposite angles of the triangle.)

$\angle cbd = \angle adb+\angle a$

$\Rightarow \angle a =62.49-33 =29.49^\circ$

Now, let us consider the right angled $\triangle acd$.

We have the value of $\angle a$ and perpendicular dc.

We have to find the hypotenuse ad.

Let us use the sine identity:

$sin\theta =\dfrac{Perpendicular}{Hypotenuse}\\\Rightarrow sin\angle a =\dfrac{cd}{ad}\\\Rightarrow sin(29.49^\circ) =\dfrac{384}{ad}\\\Rightarrow ad = \dfrac{384}{0.49}\\\Rightarrow \bold{ad = 783.67\ m}$

So, the answer is:

Distance between balloon and $\bold{a}$ is = 383.67 m