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Answer :
To solve the problem of finding the height of the tree, we'll use trigonometric ratios, specifically the tangent function. Here's how you can find the answer step-by-step:
1. Identify Shaina's Height: Shaina is 5 feet 6 inches tall. First, we need to convert her height entirely to inches. Since 1 foot is 12 inches, her height in inches is:
[tex]\[
(5 \times 12) + 6 = 66 \, \text{inches}
\][/tex]
2. Distance from the Tree: Shaina is standing 20 feet away from the tree. We also need to convert this distance to inches to match the units with Shaina's height:
[tex]\[
20 \times 12 = 240 \, \text{inches}
\][/tex]
3. Trigonometric Calculation: The angle of elevation from Shaina's eyes to the bird at the top of the tree is 68 degrees. Using the tangent of this angle, we can find the vertical distance from Shaina's eyes to the bird, which is effectively the height of the tree above Shaina's height.
- The formula for the tangent of an angle is:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite side}}{\text{adjacent side}}
\][/tex]
- Here, the opposite side is the height from Shaina's eyes to the top of the tree, and the adjacent side is the horizontal distance (240 inches):
[tex]\[
\text{height above Shaina's eyes} = 240 \times \tan(68^\circ)
\][/tex]
This calculation gives us approximately 594.02 inches.
4. Calculate Total Height of the Tree: To find the total height of the tree, add Shaina's height to the height from her eyes to the top of the tree:
[tex]\[
\text{Total height in inches} = 66 + 594.02 = 660.02 \, \text{inches}
\][/tex]
5. Convert to Feet: Finally, convert the total height of the tree from inches back to feet:
[tex]\[
\frac{660.02}{12} \approx 55 \, \text{feet}
\][/tex]
Therefore, the height of the tree is approximately 55 feet.
1. Identify Shaina's Height: Shaina is 5 feet 6 inches tall. First, we need to convert her height entirely to inches. Since 1 foot is 12 inches, her height in inches is:
[tex]\[
(5 \times 12) + 6 = 66 \, \text{inches}
\][/tex]
2. Distance from the Tree: Shaina is standing 20 feet away from the tree. We also need to convert this distance to inches to match the units with Shaina's height:
[tex]\[
20 \times 12 = 240 \, \text{inches}
\][/tex]
3. Trigonometric Calculation: The angle of elevation from Shaina's eyes to the bird at the top of the tree is 68 degrees. Using the tangent of this angle, we can find the vertical distance from Shaina's eyes to the bird, which is effectively the height of the tree above Shaina's height.
- The formula for the tangent of an angle is:
[tex]\[
\tan(\text{angle}) = \frac{\text{opposite side}}{\text{adjacent side}}
\][/tex]
- Here, the opposite side is the height from Shaina's eyes to the top of the tree, and the adjacent side is the horizontal distance (240 inches):
[tex]\[
\text{height above Shaina's eyes} = 240 \times \tan(68^\circ)
\][/tex]
This calculation gives us approximately 594.02 inches.
4. Calculate Total Height of the Tree: To find the total height of the tree, add Shaina's height to the height from her eyes to the top of the tree:
[tex]\[
\text{Total height in inches} = 66 + 594.02 = 660.02 \, \text{inches}
\][/tex]
5. Convert to Feet: Finally, convert the total height of the tree from inches back to feet:
[tex]\[
\frac{660.02}{12} \approx 55 \, \text{feet}
\][/tex]
Therefore, the height of the tree is approximately 55 feet.
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