To determine the length of the shadow of a tree that is 45 feet tall and casts a shadow forming an angle of 54° with the ground, we can use trigonometry, specifically the tangent function.
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In this scenario:
- The opposite side is the height of the tree ([tex]\( 45 \)[/tex] feet).
- The adjacent side is the length of the shadow, which we need to find.
- The angle between the shadow and the ground is [tex]\( 54^\circ \)[/tex].
The tangent function is represented as:
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Here, [tex]\( \theta = 54^\circ \)[/tex], the opposite side is the height of the tree (45 feet), and the adjacent side is the shadow length ([tex]\( x \)[/tex]).
We can rearrange the tangent function to solve for [tex]\( x \)[/tex]:
[tex]\[ \tan(54^\circ) = \frac{45}{x} \][/tex]
This can be rearranged to:
[tex]\[ x = \frac{45}{\tan(54^\circ)} \][/tex]
Now, let's calculate [tex]\( \tan(54^\circ) \)[/tex]:
[tex]\[ \tan(54^\circ) \approx 1.37638 \][/tex]
Using this value, we can find [tex]\( x \)[/tex]:
[tex]\[ x = \frac{45}{1.37638} \approx 32.70 \][/tex]
Therefore, the length of the shadow is approximately [tex]\( 32.70 \)[/tex] feet, rounded to the nearest hundredth.
