Question: Consider the following figure.
FIGURE
3 ft
4 1/2 ft
2.1 ft
\theta
Figure Description
Determine the angle $\theta$ (in degrees) in the design of the streetlight shown in the figure, where the height is $2.1$ ft. (Round your answer to one decimal place.)
$\theta = \degree$
1. **Problem Statement:** We need to find the angle $\theta$ formed between the vertical pole and the upper arm of the streetlight.
2. **Given:**
- Height of the pole (vertical side) = $2.1$ ft
- Length of the upper arm (hypotenuse) = $3$ ft
- Length of the lower diagonal support = $4 \frac{1}{2} = 4.5$ ft (not directly needed for $\theta$)
3. **Formula:** To find angle $\theta$ in a right triangle, use the cosine function:
$$\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$
Here, the adjacent side to $\theta$ is the vertical pole height $2.1$ ft, and the hypotenuse is the upper arm $3$ ft.
4. **Calculate cosine of $\theta$:**
$$\cos(\theta) = \frac{2.1}{3}$$
5. **Simplify the fraction:**
$$\cos(\theta) = \frac{\cancel{2.1}}{\cancel{3}} = 0.7$$
6. **Find $\theta$ by taking the inverse cosine (arccos):**
$$\theta = \cos^{-1}(0.7)$$
7. **Calculate $\theta$ using a calculator:**
$$\theta \approx 45.57^\circ$$
8. **Round to one decimal place:**
$$\theta \approx 45.6^\circ$$
**Final answer:**
$$\boxed{\theta = 45.6^\circ}$$