Chapter 1, Problem 48P

(a)

To determine

Draw the picture of indicating the relationship between the mountain and the distances of woman in different locations.

Explanation of Solution

Solution:

The figure representing the relationship between the height of mountain and the elevation of mountain top from different distances of woman in different locations is,

Explanation:

Given info:

The woman measures the mountain top at an elevation of $12.0°$ from the initial position at the distance x. After moving 1.00 km closer to the mountain she measures the mountain top at an elevation of $14.0°$. The height of the mountain is y.

Thus, following figure shows the relationship between the mountain and the distances of woman in different locations.

Conclusion:

The figure representing the relationship between the height of mountain and the elevation of mountain top from different distances of woman in different locations is,

(b)

To determine

Express the elevations and distances in terms of variables and mark on the figure.

Explanation of Solution

Solution:

The figure representing the relationship between the height of mountain and the elevation of mountain top from different distances of woman in different locations after replacing with variables is,

Explanation:

Given info:

The woman measures the mountain top at an elevation of $\theta$ from the initial position at the distance x. After moving a distance d closer to the mountain she measures the mountain top at an elevation of $\varphi$. The height of the mountain is y.

Thus, figure representing the relationship between the height of mountain and the elevation of mountain top from different distances of woman in different locations after replacing with variables is,

Conclusion:

The figure representing the relationship between the height of mountain and the elevation of mountain top from different distances of woman in different locations after replacing with variables is,

(c)

To determine

Trigonometric relations between the elevations and distances and the height of mountain.

Answer to Problem 48P

Solution:

The trigonometric relations between the elevations and distances and the height are $\tan 12.0°=\frac{y}{x}$ and $\tan 14.0°=\frac{y}{x-1.00\text{km}}$.

Explanation of Solution

Given info:

Expression for the relationship between the elevation of mountain’s height with the initial position of the woman, the height of the mountain and the initial distance of the woman is,

$\tan \theta =\frac{y}{x}$

• $\theta$ is the elevation of mountain’s height with the initial position of the woman
• $y$ is the height of the mountain
• $x$ is the initial distance of the woman

Substitute $12.0°$ for $\theta$ to rewrite the equation.

$\tan 12.0°=\frac{y}{x}$

Expression for the relationship between the elevation of mountain’s height with the final position of the woman, the height of the mountain and the change in distance of the woman is,

$\tan \varphi =\frac{y}{x-d}$

• $\varphi$ is the elevation of mountain’s height with the final position of the woman
• $d$ is the change in distance

Substitute $14.0°$ for $\varphi$ and  $1.00\text{km}$ for $d$ to rewrite the equation.

$\tan 14.0°=\frac{y}{x-\left(1.00\text{km}\right)}$

Thus, the trigonometric relations between the elevations and distances and the height are $\tan 12.0°=\frac{y}{x}$ and $\tan 14.0°=\frac{y}{x-1.00\text{km}}$.

Conclusion:

The trigonometric relations between the elevations and distances and the height are $\tan 12.0°=\frac{y}{x}$ and $\tan 14.0°=\frac{y}{x-1.00\text{km}}$.

(d)

To determine

The height of mountain.

Answer to Problem 48P

Solution:

The height of mountain is $1.44\times {10}^3\text{m}$.

Explanation of Solution

Expression for the relationship between the elevation of mountain’s height with the initial position of the woman, the height of the mountain and the initial distance of the woman is,

$\tan \theta =\frac{y}{x}$

Rewrite the equation in terms x.

$x=\frac{y}{\tan \theta }$

Expression for the relationship between the elevation of mountain’s height with the final position of the woman, the height of the mountain and the change in distance of the woman is,

$\begin{array}{c}\tan \varphi =\frac{y}{x-d}\\ =\frac{y}{\frac{y}{\tan \theta }-d}\\ =\frac{y\cdot \tan \theta }{y-d\cdot \tan \theta }\end{array}$

Rewrite the equation in terms of y.

$y=\frac{d\cdot \tan \theta \cdot \tan \varphi }{\tan \varphi -\tan \theta }$

Substitute $12.0°$ for $\theta$, $14.0°$ for $\varphi$ and  $1.00\text{km}$ for $d$ to find y.

$\begin{array}{c}y=\frac{\left(1.00\text{km}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)\cdot \tan 12.0°\cdot \tan 14.0°}{\tan 14.0°-\tan 12.0°}\\ =1.44\times {10}^3\text{m}\end{array}$

Conclusion:

Therefore, the height of mountain is $1.44\times {10}^3\text{m}$