To find the height of the tree, we can break the problem down using trigonometric relationships.
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Convert Shaina's height to feet: Shaina is 5 feet, 6 inches tall. There are 12 inches in a foot, so: \[ 6 \text{ inches} = \frac{6}{12} = 0.5 \text{ feet} \] Therefore, Shaina's height in feet is: \[ 5.5 \text{ feet} \]
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Set up the problem: We have the following information:
- Shaina's height = 5.5 feet
- Distance from the base of the tree = 20 feet
- Angle of elevation to the woodpecker = 68 degrees
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Use the tangent function to find the height of the tree above Shaina's head: The height of the tree above Shaina's head is found using the tangent of the angle of elevation: \[ \tan(68^\circ) = \frac{\text{height above Shaina}}{\text{distance from tree}} = \frac{\text{height above Shaina}}{20 \text{ feet}} \]
Rearranging the formula to solve for the height above Shaina: \[ \text{height above Shaina} = 20 \cdot \tan(68^\circ) \]
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Calculate the height above Shaina: Now we can calculate \(\tan(68^\circ)\): \[ \tan(68^\circ) \approx 2.475 \] Therefore: \[ \text{height above Shaina} \approx 20 \cdot 2.475 \approx 49.5 \text{ feet} \]
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Calculate the total height of the tree: To find the total height of the tree, add Shaina's height to the height above her: \[ \text{Total height of the tree} = \text{Shaina's height} + \text{height above Shaina} = 5.5 + 49.5 = 55 \text{ feet} \]
Since 55 feet does not match with either of the responses, let's check the calculations again:
- Confirming tangent calculation using a calculator: \(\tan(68^\circ) \approx 2.475 \text{ (which is approximate and needs to be checked)}\).
Now, ensuring everything considered:
- Height of the tree should be confirmed therefore: \[ \text{Total height of the tree} \approx 55 + 20 \cdot \tan(68) \]
But there are no feasible options but to double-check. Thus, I'll stick to calculations confirming Shaina's height addition to the above part creating interest to check references or even find if closer holds otherwise.
Going through presented options, simplified verification might yield different result placements or could be misassigning angles/elevation affecting both via assumptions addressing potential overlook to resolve.
By this run-through, near elimination or request of found projection might point to resolving besides misunderstanding expectations.
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