To find the distance from the top of the tree to the end of the shadow, we can use trigonometry. We will consider the situation as a right triangle where:
- The distance of the shadow (22 m) is the adjacent side.
- The height of the tree (which we want to find) is the opposite side.
- The distance from the top of the tree to the end of the shadow will be the hypotenuse.
Let \( h \) be the height of the tree. The angle of elevation from the end of the shadow to the top of the tree is given as \( 83^\circ \).
We can use the following trigonometric relationships:
- The tangent of the angle relates the opposite side (height of the tree) to the adjacent side (length of the shadow): \[ \tan(83^\circ) = \frac{h}{22} \]
We can rearrange this to find \( h \): \[ h = 22 \cdot \tan(83^\circ) \]
- To find the hypotenuse \( d \) (the distance from the end of the shadow to the top of the tree), we can use the cosine function: \[ \cos(83^\circ) = \frac{22}{d} \]
Rearranging for \( d \): \[ d = \frac{22}{\cos(83^\circ)} \]
Now we can calculate \( h \) and \( d \).
Step 1: Calculate \( h \)
Using a calculator to find \( \tan(83^\circ) \): \[ \tan(83^\circ) \approx 9.514 \] So, \[ h \approx 22 \cdot 9.514 \approx 209.308 \text{ meters} \]
Step 2: Calculate \( d \)
Now we find \( d \): Using a calculator to find \( \cos(83^\circ) \): \[ \cos(83^\circ) \approx 0.094 \] So, \[ d \approx \frac{22}{0.094} \approx 234.042 \text{ meters} \]
Thus, the distance from the top of the tree to the end of the shadow is approximately: \[ \boxed{234.04 \text{ meters}} \]
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