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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

1 Answer

4 votes

Answer:

Distance between balloon and a is = 383.67 m

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 = (Perpendicular)/(Base)


tan\angle cbd =(cd)/(cb)\\\Rightarrowtan\angle cbd =(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 =(Perpendicular)/(Hypotenuse)\\\Rightarrow sin\angle a =(cd)/(ad)\\\Rightarrow sin(29.49^\circ) =(384)/(ad)\\\Rightarrow ad = (384)/(0.49)\\\Rightarrow \bold{ad = 783.67\ m}

So, the answer is:

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

three people are watching a hot air balloon travel over their town. at a certain point-example-1
User Ganzolo
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