Question

The diagram shows the sculpture Clothespin casting a shadow. In the diagram, $\angle 2$is the angle of the sun above the horizon.

1. At a certain time of the day, $m\angle 1=148°$, and the shadow has a length of 72 feet. Use a calculator to find the height of the sculpture. Round to the nearest tenth.
2. At a certain time of the day, $m\angle 1=135°$. How is the length of the shadow related to the height of the sculpture? Explain your reasoning.

Short answer

1. The height of the sculpture is $45.0$ feet.
2. The length of shadow is equal to the height of the sculpture.

Step-by-step solution

a.Step 1. Given Information.

The figure is shown below.

Step 2. Concept used.

Sum of the linear pairs of angles is $180°$.

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

Step 3. Calculation.

Since, sum of the linear pairs of angles is $180°$therefore,

$\begin{array}{c}m\angle 1+\angle 2=180°\\ {148}^{\circ }+\angle 2=180°\\ \angle 2=180°-148°\\ \angle 2=32°\end{array}$

Also,

$\begin{array}{c}\tan \theta =\frac{\text{perpendicular}}{\text{base}}\\ \tan \theta =\frac{\text{heightofsculpture}}{\text{lengthofshadow}}\\ \tan \theta =\frac{\text{heightofsculpture}}{\text{lengthofshadow}}\\ \text{heightofsculpture}=\tan \theta \times \text{lengthofshadow}\end{array}$

Since, $\theta =\angle 2=32°$.

Therefore,

$\begin{array}{c}\text{heightofsculpture}=\tan 32°\times 72\\ =0\cdot 6248\times 72\\ \approx 44\cdot 9906\\ \approx 45.0\end{array}$

Thus, the height of the sculpture round to the nearest tenth is $45.0$feet.

Step 4. Conclusion.

At certain time of the day $m\angle 1=148°$ and length of the shadow is 72 feet, the height of the sculpture round to the nearest tenth is $45.0\text{feet}$.

b.Step 1. Given Information.

The figure is shown below.

Step 2. Concept used.

Sum of the linear pairs of angles is $180°$.

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

Step 3. Calculation.

Since, sum of the linear pairs of angles is $180°$ therefore,

$\begin{array}{c}m\angle 1+\angle 2=180°\\ {135}^{\circ }+\angle 2=180°\\ \angle 2=180°-135°\\ \angle 2=45°\end{array}$

Also,

$\begin{array}{c}\tan \theta =\frac{\text{perpendicular}}{\text{base}}\\ \tan \theta =\frac{\text{heightofsculpture}}{\text{lengthofshadow}}\\ \tan \theta =\frac{\text{heightofsculpture}}{\text{lengthofshadow}}\\ \text{heightofsculpture}=\tan \theta \times \text{lengthofshadow}\end{array}$

Since, $\theta =\angle 2=45°$.

Therefore,

$\begin{array}{c}\text{heightofsculpture}=\tan 45°\times \text{lengthofshadow}\\ =1\times \text{lengthofshadow}\\ =\text{lengthofshadow}\end{array}$

Step 4. Conclusion.

At certain time of the day $m\angle 1=135°$ and length of the shadow is equal to the height of the sculpture.