To find the height of the tree, we’ll first convert Shaina’s height from feet and inches to inches only. Shaina is 5 feet 6 inches tall, which is:
\[ 5 \text{ feet} \times 12 \text{ inches/foot} + 6 \text{ inches} = 60 \text{ inches} + 6 \text{ inches} = 66 \text{ inches} \]
Next, we can set up the problem. Shaina is standing 20 feet away from the base of the tree, and we need to find the height of the tree at an angle of elevation of 68 degrees.
The total height of the tree \( h_{\text{tree}} \) can be calculated using the tangent function in the right triangle formed by Shaina's position, the top of the tree, and the base of the tree.
First, let's convert the distance from feet to inches (since we are measuring height in inches): \[ 20 \text{ feet} = 20 \times 12 = 240 \text{ inches} \]
Now, we can use the tangent function from trigonometry: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
The opposite side is the height from Shaina's eyes to the top of the tree, and adjacent side is the distance from her to the tree: \[ \tan(68^\circ) = \frac{h - 66}{240} \]
Rearranging gives: \[ h - 66 = 240 \tan(68^\circ) \]
Calculating \(\tan(68^\circ)\): \[ \tan(68^\circ) \approx 2.4751 \] Then substitute that value in: \[ h - 66 \approx 240 \times 2.4751 \approx 594.24 \text{ inches} \] Thus: \[ h \approx 594.24 + 66 = 660.24 \text{ inches} \]
Therefore, the total height of the tree is approximately 660.24 inches. Now, we can choose the closest response from the provided options:
- 240 inches
- 680 inches
- 66 feet (which is 792 inches)
- 55 feet (which is 660 inches)
Thus, since 660 inches is closest to 680 inches, the answer is:
680 inches
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