Chapter 12.8, Problem 14P

To determine

Find the distance between balloons and which balloon is higher and by how much.

Answer to Problem 14P

Distance between balloons = 1.897 km

Balloon to the left is higher than the balloon to the right by 38 meters.

Explanation of Solution

Formula Used:

  $hypotenuse\cdot \sin \theta =\text{length of side opposite to }\theta$

  $hypotenuse\cdot \cos \theta =\text{length of side adjacent to }\theta$

Law of Sine:

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

  

Calculation:

Represent the situation and solve :

All the angles can be calculated by the properties of linear angles and sum of all the angles of a triangle is $180°$ .

  

Use Law of Sine:

  $\begin{array}{l}\frac{1}{\sin 23°}=\frac{z}{\sin 19°}\\ \Rightarrow z=\sin 19°\cdot \frac{1}{\sin 23°}\mathrm{.................}\text{[multiply each side by }\sin 19°]\\ \Rightarrow z\approx 0.833\text{ km = 833 m }\end{array}$

Again , use law of sine:

  $\begin{array}{l}\frac{1}{\sin 40°}=\frac{r}{\sin 25°}\\ \Rightarrow r=\sin 25°\cdot \frac{1}{\sin 40°}\mathrm{.................}\text{[multiply each side by }\sin 25°]\\ \Rightarrow r\approx 0.657\text{ km = 657 m }\end{array}$

Now,

  $\begin{array}{l}hypotenuse\cdot \cos \theta =\text{length of side adjacent to }\theta \\ \Rightarrow z\cos 42°=x\\ \Rightarrow x=0.833(0.743)\\ \Rightarrow x\approx 0.619\text{ km}\end{array}$

  $\begin{array}{l}hypotenuse\cdot \cos \theta =\text{length of side adjacent to }\theta \\ \Rightarrow r\cos 65°=p\\ \Rightarrow p=0.657(0.4226)\\ \Rightarrow p\approx 0.278\text{ km}\end{array}$

So, the distance between the two balloons is :

  $p+1+x\approx 0.278+1+0.619=1.897\text{ km}$

Now we find the heights of the balloons:

  $\begin{array}{l}hypotenuse\cdot \sin \theta =\text{length of side opposite to }\theta \\ \Rightarrow z\sin 42°=y\\ \Rightarrow y=(0.833)(0.669)\\ \Rightarrow y\approx 0.557\text{ km = 557 m}\end{array}$

  $\begin{array}{l}hypotenuse\cdot \sin \theta =\text{length of side opposite to }\theta \\ \Rightarrow r\sin 65°=q\\ \Rightarrow q=(0.657)(0.906)\\ \Rightarrow q\approx 0.595\text{ km = 595 m}\end{array}$

So, the height of the balloon to the left is more.

It is more than the balloon to the right by 595-557 = 38 meters.