This image presents a classic right-triangle trigonometry problem, likely related to finding the length of a shadow or the distance to an object based on its height and the angle of elevation of the sun.
The diagram shows a right triangle with:
- One acute angle labeled as 22°.
- The side opposite this angle is labeled as 60m (this represents the height of an object, like a tree or a building).
- The right angle is at the base, indicating the ground is perpendicular to the object.
- The hypotenuse represents the line from the top of the object to the sun (or more accurately, the direction of the sunlight).
The most logical question implied by this diagram is:
What is the length of the shadow cast by the object?
This corresponds to the side adjacent to the 22° angle. We can solve for this using the tangent function, which relates the opposite and adjacent sides of a right triangle.
The formula is:
$$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} $$
In this case:
$$ \tan(22^\circ) = \frac{60}{\text{shadow length}} $$
To find the shadow length, we rearrange the formula:
$$ \text{shadow length} = \frac{60}{\tan(22^\circ)} $$
I will now calculate this value.
```py
import math
# Given values
opposite_side = 60 # meters
angle_degrees = 22
# Convert angle to radians for calculation
angle_radians = math.radians(angle_degrees)
# Calculate the adjacent side (shadow length)
shadow_length = opposite_side / math.tan(angle_radians)
shadow_length
```
```py
148.50718970836773
```
Based on the calculation, the length of the shadow is approximately 148.5 meters.
---
Solution:
The problem in the image is to find the length of the shadow cast by an object that is 60 meters tall when the sun's rays hit the ground at an angle of 22 degrees.
Step-by-step solution:
1.
Identify the knowns: We have a right triangle where:
- The angle of elevation of the sun is $22^\circ$.
- The height of the object (the side opposite the angle) is 60 meters.
- We need to find the length of the shadow (the side adjacent to the angle).
2.
Choose the correct trigonometric ratio: The tangent function relates the opposite side to the adjacent side.
$$ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} $$
3.
Set up the equation:
$$ \tan(22^\circ) = \frac{60}{\text{shadow length}} $$
4.
Solve for the unknown:
$$ \text{shadow length} = \frac{60}{\tan(22^\circ)} $$
5.
Calculate the result:
Using a calculator, $\tan(22^\circ) \approx 0.4040$.
$$ \text{shadow length} = \frac{60}{0.4040} \approx 148.5 \text{ meters} $$
Therefore, the length of the shadow is
148.5 meters.
Parent Tip: Review the logic above to help your child master the concept of trig word problems worksheet answer key.
