Chapter 10.PS, Problem 1PS

Mounting Climbing Several mountain climbers are located in a mountain pass between two peaks. The angles of elevation to the two peaks are $0.84$ radian and $1.10$ radians. A range finder shows that the distance to the peaks are $3250$ feet and $6700$ feet, respectively (see figure).

(a) Find the angle between the two lines.

(b) Approximate the amount of vertical climb that is necessary to reach the summit of each peak.

To determine

(a)

To find:

The angle between the two lines.

Answer to Problem 1PS

Solution:

The angle between the two lines is $1.2016\text{radians}$.

Explanation of Solution

Given:

The angle of elevation to the two peaks are $0.84\text{radian}$ and $1.10\text{radians}$

The given diagram is shown in figure $\left(1\right)$.

Figure $\left(1\right)$

Approach:

The sum of the angle of elevations of the two peaks and the angle between the sight to the peaks is equal to $\pi \text{radians}$.

Calculation:

Suppose the angle between two lines of sight to the peaks be $\theta$ and it is given that the angles of elevation to the two peaks are $0.84\text{radian}$ and $1.10\text{radians}$.

So the total of the all angles must be equal to $\pi \text{radians}$.

$\begin{array}{rll}0.84+1.10+\theta & =& \pi \\ \theta & =& \pi -1.94\\ \theta & =& 3.14-1.94\\ \theta & =& 1.2016\text{radians}\end{array}$

Therefore, the angle between the two lines is $1.2016\text{radians}$.

To determine

(b)

To find:

The Approximate amount of vertical climb that is necessary to reach the summit of each peak.

Answer to Problem 1PS

Solution:

The Approximate amount of vertical climb to the peak is $2420\text{feets}$ and $5971\text{feets}$.

Explanation of Solution

Approach:

The formula of $\sin \theta =\frac{P}{H}$.

Here, $P$ is the altitude and $H$ is the elevation of any summit.

And the value of $\text{radians}$ in $\text{sine}$ form is,

$\begin{array}{l}\sin 0.84=0.7446\\ \sin 1.10=0.8912\end{array}$

Calculation:

Suppose the vertical climb for the first peak is $x$ and $\theta =0.84$. Then,

$\begin{array}{l}\sin 0.86=\frac{x}{3250}\\ x=3250\times 0.7446\\ x=2420\text{feet}\end{array}$

Consider, the vertical climb for the second peak be $y$ and $\theta =1.10$. Then,

$\begin{array}{l}\sin 01.10=\frac{y}{6700}\\ y=6700\times 0.8912\\ y=5971\text{feet}\end{array}$

Therefore, the approximate amount of vertical climb to the peak is $2420\text{feets}$ and $5971\text{feets}$.