Chapter 4.8, Problem 37E

a.

To determine

To find: The height of the sundial.

Answer to Problem 37E

Length $\text{= 161}\text{.26 ft}\text{.}$

Explanation of Solution

Given information: Length $\text{= 217 ft}\text{.}$

And angle $={42}^{\circ }$

Calculation:

  

Let, $h$ be the height of the sundial.

Here,

  $\begin{array}{l}\text{AC}=217\text{ ft}\text{.}\\ \&\angle \text{CAP}={42}^{\circ }\end{array}$

Find $\angle \text{CAB}$

  $\angle \text{CAB}={90}^{\circ }-{42}^{\circ }={48}^{\circ }$

Now,

  $\sin \theta =\frac{\text{opp}}{\text{hyp}}$

From $\Delta \text{ABC}$

  $\begin{array}{l}\sin {48}^{\circ }=\frac{h}{217}\\ \Rightarrow h=217\sin {48}^{\circ }\\ \Rightarrow h\simeq 161.26\text{ ft}\text{.}\end{array}$

b.

To determine

To find: The height of the sundial.

Answer to Problem 37E

  $d$ is at right angle to $h$ and angle of elevation to sun is $\theta$ as shown in figure.

Explanation of Solution

Given information: Length $=217\text{ ft}.$

And angle $={42}^{\circ }$

Calculation:

  

Let, $h$ be the height of the sundial.

Here,

  $\begin{array}{l}\text{AC}=217\text{ ft}\text{.}\\ \&\angle \text{CAP}={42}^{\circ }\end{array}$

Find $\angle \text{CAB}$

  $\angle \text{CAB}={90}^{\circ }-{42}^{\circ }={48}^{\circ }$

  $d$ is at right angle to $h$ and angle of elevation to sun is $\theta$ as shown in figure.

  

c.

To determine

To find: The height of the sundial.

Answer to Problem 37E

  $d=\frac{161.26}{\tan \theta }$

Explanation of Solution

Given information: Length $=217\text{ ft}.$

And angle $={42}^{\circ }$

Calculation:

  

Let, $h$ be the height of the sundial.

Here,

  $\begin{array}{l}\text{AC}=217\text{ ft}\text{.}\\ \&\angle \text{CAP}={42}^{\circ }\end{array}$

Find $\angle \text{CAB}$

  $\angle \text{CAB}={90}^{\circ }-{42}^{\circ }={48}^{\circ }$

  $\begin{array}{l}\tan \theta =\frac{h}{d}\\ \Rightarrow d=\frac{h}{\tan \theta }\\ \Rightarrow d=\frac{161.26}{\tan \theta }\end{array}$

d.

To determine

To find: The height of the sundial.

Answer to Problem 37E

  $\begin{array}{l}d=\frac{161.26}{\tan {10}^{\circ }}\left[at\theta {\text{=10}}^{\circ }\right]\\ \Rightarrow d\simeq 914.56\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {20}^{\circ }}\left[at\theta {\text{=20}}^{\circ }\right]\\ \Rightarrow d\simeq 443.06ft.\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {30}^{\circ }}\left[at\theta {\text{=30}}^{\circ }\right]\\ \Rightarrow d\simeq 279.3\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {40}^{\circ }}\left[at\theta {\text{=40}}^{\circ }\right]\\ \Rightarrow d\simeq 192.18\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {50}^{\circ }}\left[at\theta {\text{=50}}^{\circ }\right]\\ \Rightarrow d\simeq 135.31\text{ ft}\text{.}\end{array}$

Explanation of Solution

Given information: Length $=217\text{ ft}.$

And angle $={42}^{\circ }$

Calculation:

  

Let, $h$ be the height of the sundial.

Here,

  $\begin{array}{l}\text{AC}=217\text{ ft}\text{.}\\ \&\angle \text{CAP}={42}^{\circ }\end{array}$

It has to find $\angle \text{CAB}$

  $\angle \text{CAB}={90}^{\circ }-{42}^{\circ }={48}^{\circ }$

  $\begin{array}{l}d=\frac{161.26}{\tan {10}^{\circ }}\left[at\theta {\text{=10}}^{\circ }\right]\\ \Rightarrow d\simeq 914.56\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {20}^{\circ }}\left[at\theta {\text{=20}}^{\circ }\right]\\ \Rightarrow d\simeq 443.06\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {30}^{\circ }}\left[at\theta {\text{=30}}^{\circ }\right]\\ \Rightarrow d\simeq 279.3\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {40}^{\circ }}\left[at\theta {\text{=40}}^{\circ }\right]\\ \Rightarrow d\simeq 192.18\text{ ft}\text{.}\end{array}$

  $\begin{array}{l}d=\frac{161.26}{\tan {50}^{\circ }}\left[at\theta {\text{=50}}^{\circ }\right]\\ \Rightarrow d\simeq 135.31\text{ ft}\text{.}\end{array}$

e.

To determine

To find: The height of the sundial.

Answer to Problem 37E

  $d$ Decrease with increase the value of $\theta$ .

Explanation of Solution

Given information: Length $=217\text{ ft}.$

And angle $={42}^{\circ }$

Calculation:

  

Let, $h$ be the height of the sundial.

Here,

  $\begin{array}{l}\text{AC}=217\text{ ft}\text{.}\\ \&\angle \text{CAP}={42}^{\circ }\end{array}$

It has to find $\angle \text{CAB}$

  $\angle \text{CAB}={90}^{\circ }-{42}^{\circ }={48}^{\circ }$

  $d$ Decrease with increase the value of $\theta$ .