Question

A tree casts a shadow that is 22 meters long. The angle of elevation from the end of the shadow to the top of the tree is 83°. Find the distance from the top of the tree to the end of the shadow.

Answer 1

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:

1. 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 \( h \):
\[
h = 22 \cdot \tan(83^\circ)
\]

2. 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:
\[
\boxed{234.04 \text{ meters}}
\]