Certainly! Let's work through this problem step-by-step to determine the angle of elevation at which the archer should shoot in order to hit the target.
### Step-by-Step Solution:
1. Identify the Given Data:
- Archer's height (hâ): 1.5 meters
- Target's height (hâ): 4.5 meters
- Horizontal distance between target and archer (d): 3 meters
2. Calculate the Height Difference:
- The height difference between the archer and the target ([tex]\(\Delta h\)[/tex]):
[tex]\[
\Delta h = h_2 - h_1
\][/tex]
Substituting the given values:
[tex]\[
\Delta h = 4.5 \, \text{meters} - 1.5 \, \text{meters} = 3.0 \, \text{meters}
\][/tex]
3. Use Trigonometry to Determine the Angle of Elevation:
- We employ the tangent function, which is the ratio of the opposite side (height difference) to the adjacent side (horizontal distance):
[tex]\[
\tan(\theta) = \frac{\Delta h}{d}
\][/tex]
[tex]\[
\tan(\theta) = \frac{3.0 \, \text{meters}}{3 \, \text{meters}} = 1
\][/tex]
4. Calculate the Angle:
- To find the angle [tex]\(\theta\)[/tex], we take the inverse tangent (arctan) of 1:
[tex]\[
\theta = \arctan(1)
\][/tex]
5. Convert from Radians to Degrees (if necessary):
- The inverse tangent of 1 is [tex]\(\frac{\pi}{4}\)[/tex] radians.
- Converting [tex]\(\frac{\pi}{4}\)[/tex] radians to degrees:
[tex]\[
\theta = 45^\circ
\][/tex]
### Conclusion:
The angle of elevation at which the archer should shoot the arrow to hit the target is 45°. Therefore, the correct answer is:
(B) 45°
