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Answer :
Sure, let's solve the problem step-by-step using trigonometric ratios.
1. Convert Shaina's height to feet:
- Shaina's height is 5 feet and 6 inches.
- We first convert her height to feet. Since there are 12 inches in a foot:
[tex]\[
5 \text{ feet} + \frac{6 \text{ inches}}{12} = 5.5 \text{ feet}
\][/tex]
2. Understand the given information:
- Angle of elevation to the top of the tree: [tex]\(68^\circ\)[/tex]
- Distance from Shaina to the base of the tree: 20 feet
- Shaina's height: 5.5 feet
3. Set up the trigonometric ratio:
- We will use the tangent function. Tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, the "opposite" side is the height from Shaina's eyes to the top of the tree, and the "adjacent" side is the distance from Shaina to the base of the tree.
4. Calculate the height from Shaina's eyes to the top of the tree:
- Let [tex]\( h \)[/tex] be the height from Shaina's eyes to the top of the tree.
[tex]\[
\tan(68^\circ) = \frac{h}{20 \text{ feet}}
\][/tex]
- Solving for [tex]\( h \)[/tex]:
[tex]\[
h = 20 \times \tan(68^\circ)
\][/tex]
5. Calculate the total height of the tree:
- We need to add Shaina's height to the height from her eyes to the top of the tree to get the total height of the tree:
[tex]\[
\text{Total height of the tree} = h + 5.5 \text{ feet}
\][/tex]
Using the given result:
[tex]\[ \tan(68^\circ) \approx 2.4751 \][/tex]
So:
[tex]\[ h = 20 \times 2.4751 \approx 49.5 \text{ feet} \][/tex]
Adding Shaina's height:
[tex]\[ \text{Total height of the tree} = 49.5 + 5.5 = 55 \text{ feet} \][/tex]
Therefore, the height of the tree is:
[tex]\[
\boxed{55 \text{ feet}}
\][/tex]
1. Convert Shaina's height to feet:
- Shaina's height is 5 feet and 6 inches.
- We first convert her height to feet. Since there are 12 inches in a foot:
[tex]\[
5 \text{ feet} + \frac{6 \text{ inches}}{12} = 5.5 \text{ feet}
\][/tex]
2. Understand the given information:
- Angle of elevation to the top of the tree: [tex]\(68^\circ\)[/tex]
- Distance from Shaina to the base of the tree: 20 feet
- Shaina's height: 5.5 feet
3. Set up the trigonometric ratio:
- We will use the tangent function. Tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
- Here, the "opposite" side is the height from Shaina's eyes to the top of the tree, and the "adjacent" side is the distance from Shaina to the base of the tree.
4. Calculate the height from Shaina's eyes to the top of the tree:
- Let [tex]\( h \)[/tex] be the height from Shaina's eyes to the top of the tree.
[tex]\[
\tan(68^\circ) = \frac{h}{20 \text{ feet}}
\][/tex]
- Solving for [tex]\( h \)[/tex]:
[tex]\[
h = 20 \times \tan(68^\circ)
\][/tex]
5. Calculate the total height of the tree:
- We need to add Shaina's height to the height from her eyes to the top of the tree to get the total height of the tree:
[tex]\[
\text{Total height of the tree} = h + 5.5 \text{ feet}
\][/tex]
Using the given result:
[tex]\[ \tan(68^\circ) \approx 2.4751 \][/tex]
So:
[tex]\[ h = 20 \times 2.4751 \approx 49.5 \text{ feet} \][/tex]
Adding Shaina's height:
[tex]\[ \text{Total height of the tree} = 49.5 + 5.5 = 55 \text{ feet} \][/tex]
Therefore, the height of the tree is:
[tex]\[
\boxed{55 \text{ feet}}
\][/tex]
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