Chapter 12.1, Problem 32E

a.

To determine

To find: The height of the sculpture round to the nearest tenth.

Answer to Problem 32E

The height of the sculpture round to the nearest tenth is $64\cdot 829$ .

Explanation of Solution

Given:

  $\angle 2$ is the angle of the sun above the horizon. At certain time of the day

  $m\angle 1={148}^{\circ }$ length of the shadow is $72$ feet.

Concept used:

Sum of the linear pairs of angles is ${180}^{\circ }$ .

  $\tan \theta =\frac{perpendicular}{base}$

Calculation:

Since, sum of the linear pairs of angles is ${180}^{\circ }$ .

Therefore,

  $m\angle 1+\angle 2={180}^{\circ }$

  $\begin{array}{l}\Rightarrow {148}^{\circ }+\angle 2={180}^{\circ }\\ \Rightarrow \angle 2={180}^{\circ }-{148}^{\circ }\\ \Rightarrow \angle 2={42}^{\circ }\end{array}$

Also,

  $\tan \theta =\frac{perpendicular}{base}$

  $\begin{array}{l}\Rightarrow \tan \theta =\frac{\text{height of sculpture}}{\text{length of shadow}}\\ \Rightarrow \tan \theta =\frac{\text{height of sculpture}}{\text{length of shadow}}\\ \Rightarrow \text{height of sculpture}=\tan \theta \times \text{length of shadow}\end{array}$

Since,

  $\theta =\angle 2={42}^{\circ }$

Therefore,

  $\begin{array}{l}\Rightarrow \text{height of sculpture}=\tan {42}^{\circ }\times 72\\ \Rightarrow \text{height of sculpture}=0\cdot 9004\times 72\\ \Rightarrow \text{height of sculpture}=64\cdot 829\end{array}$

Thus, the height of the sculpture round to the nearest tenth is $64\cdot 829$ .

Conclusion:

At certain time of the day

  $m\angle 1={148}^{\circ }$ and length of the shadow is $72$ feet, the height of the sculpture round to the nearest tenth is $64\cdot 829$ .

b.

To determine

To show: How length of the shadow is related to the height of the sculpture.

Answer to Problem 32E

Length of the shadow = Height of the shadow.

Explanation of Solution

Given:

  $\angle 2$ is the angle of the sun above the horizon. At certain time of the day

  $m\angle 1={135}^{\circ }$ .

Concept Used:

Sum of the linear pairs of angles is ${180}^{\circ }$ .

  $\tan \theta =\frac{perpendicular}{base}$

Calculation:

Since, sum of the linear pairs of angles is ${180}^{\circ }$ .

Therefore,

  $m\angle 1+\angle 2={180}^{\circ }$

  $\begin{array}{l}\Rightarrow {135}^{\circ }+\angle 2={180}^{\circ }\\ \Rightarrow \angle 2={180}^{\circ }-{135}^{\circ }\\ \Rightarrow \angle 2={45}^{\circ }\end{array}$

Also,

  $\tan \theta =\frac{perpendicular}{base}$

Since,

  $\theta =\angle 2={45}^{\circ }$

Therefore,

  $\begin{array}{l}\Rightarrow \tan {45}^{\circ }=\frac{\text{height of sculpture}}{\text{length of shadow}}\\ \Rightarrow 1=\frac{\text{height of sculpture}}{\text{length of shadow}}\end{array}$

  $\Rightarrow \text{height of sculpture}=\text{length of shadow}$

Hence,

  $\text{length of shadow}=\text{height of sculpture}$ .

Conclusion:

At certain time of the day when $m\angle 1={135}^{\circ }$ then $\text{length of shadow}=\text{height of sculpture}$ .