Chapter 12.8, Problem 12P

To determine

Find the distance of summit from sea level.

Answer to Problem 12P

About 6.94 km

Explanation of Solution

Given:

From Base Camp, located 2000 m above sea level, the angle of elevation of Camp A is $25°$ and the angle of elevation of the summit of Mount Snow is $41°$ .From Camp A the angle of elevation of the summit is $53°$ .The Camp A is 3 km closer to the summit (air distance) than

Base Camp is.

Calculation:

Let the distance between Base Camp and Mount Snow = x

The distance between Camp A and Base Camp = BD = 3 km

Then the distance between Camp A and Mount Snow = AC = 3 − x

Represent the situation and solve :

  

In triangle MAC ,

  $\begin{array}{l}\tan \angle MAC=\frac{\text{length of opposite side to }\angle \text{MAC}}{\text{length of adjacent side to }\angle \text{MAC}}\\ \Rightarrow \tan 53°=\frac{MC}{x-3}\\ \Rightarrow MC=(x-3)\tan 53°\mathrm{..............}[\text{multiply each side by (x-3)]}\end{array}$

In triangle ABD ,

  $\begin{array}{l}\tan \angle ABD=\frac{\text{length of opposite side to }\angle A\text{BD}}{\text{length of adjacent side to }\angle A\text{BD}}\\ \Rightarrow \tan 25°=\frac{AD}{3}\\ \Rightarrow AD=3\tan 25°\mathrm{..............}[\text{multiply each side by 3]}\end{array}$

In triangle MBE ,

  $\begin{array}{l}\tan \angle MBE=\frac{\text{length of opposite side to }\angle MBE}{\text{length of adjacent side to }\angle M\text{BE}}\\ \Rightarrow \tan 41°=\frac{ME}{x}\\ \Rightarrow ME=x\tan 41°\mathrm{..............}[\text{multiply each side by (x)]}\end{array}$

Also,

  $ME=(x-3)\tan 53°+3\tan 25°$

Equate both the values of ME:

  $\begin{array}{l}x\tan 41°=(x-3)\tan 53°+3\tan 25°\\ \Rightarrow x\tan 41°=x\tan 53°-3\tan 53°+3\tan 25°\\ \Rightarrow x\tan 41°-x\tan 53°=3(-\tan 53°+\tan 25°)\\ \Rightarrow x(\tan 41°-\tan 53°)=3(-\tan 53°+\tan 25°)\\ \Rightarrow x=\frac{3(-\tan 53°+\tan 25°)}{(\tan 41°-\tan 53°)}\mathrm{................}[\text{divide each side by }(\tan 41°-\tan 53°)]\\ \Rightarrow x\approx 5.63777\end{array}$

The height of Mount Snow from base camp is ME:

  $ME=x\tan 41°=(5.6777)\tan 41°\approx 4.936$

So, the height of Mount Snow from sea level = 2 + 4.936 = 6.94 km (approximately)