To find the height of the tree, we first need to determine Shaina's height in feet. Shaina is 5 feet, 6 inches tall, which can be converted to:
\[ 5 \text{ feet} + 6 \text{ inches} \times \frac{1 \text{ foot}}{12 \text{ inches}} = 5 + 0.5 = 5.5 \text{ feet} \]
Now, we know the following:
- The distance from Shaina to the base of the tree is 20 feet.
- The angle of elevation from Shaina's eye level to the bird is 68 degrees.
We can use trigonometry to find the height of the tree above Shaina's eye level. Using the tangent function (which is the ratio of the opposite side to the adjacent side in a right triangle):
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
Here:
- \(\theta = 68^\circ\)
- The opposite side is the height of the woodpecker above Shaina's head (let's call it \( h \)).
- The adjacent side is the distance from Shaina to the base of the tree = 20 feet.
Using the formula, we get:
\[ \tan(68^\circ) = \frac{h}{20} \]
To find \( h \), rearranging gives:
\[ h = 20 \cdot \tan(68^\circ) \]
Now, we'll calculate \( \tan(68^\circ) \):
Using a scientific calculator:
\[ \tan(68^\circ) \approx 2.4751 \]
Now substitute this back into the equation:
\[ h = 20 \cdot 2.4751 \approx 49.502 \text{ feet} \]
This is the height above Shaina's eye level. To find the total height of the tree, we add Shaina's height:
\[ \text{Height of the tree} = \text{Shaina's height} + h = 5.5 + 49.502 \approx 55.002 \text{ feet} \]
Rounding gives us approximately 55 feet.
The correct answer is:
55 feet