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Answer :
To solve the math problem using trigonometric ratios, follow these steps:
1. Understand the Problem:
- Shaina is 5 feet, 6 inches tall.
- She stands 20 feet away from the tree.
- She looks up and sees a bird at an angle of elevation of 68 degrees. We are asked to find the total height of the tree.
2. Convert Units:
- First, convert Shaina's height from feet and inches to only feet. Since she is 5 feet and 6 inches tall, convert inches into feet:
[tex]\[
6 \text{ inches} = \frac{6}{12} \text{ feet} = 0.5 \text{ feet}
\][/tex]
So, Shaina’s height is:
[tex]\[
5 + 0.5 = 5.5 \text{ feet}
\][/tex]
3. Use Trigonometric Ratios:
- The angle of elevation is formed between a horizontal line from Shaina's eye to the base of the tree and her line of sight to the bird.
- Use the tangent of the angle of elevation to find the height of the tree above Shaina’s eyes. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle.
[tex]\[
\tan(68^\circ) = \frac{\text{Height of tree above Shaina's eye}}{20}
\][/tex]
4. Solve for the Height of the Tree above Shaina's Eyes:
- Rearrange the formula to find the height above her eyes:
[tex]\[
\text{Height above eyes} = 20 \times \tan(68^\circ)
\][/tex]
- This height is approximately 49.5 feet.
5. Calculate the Total Height of the Tree:
- To find the total height of the tree, add Shaina's height to the height of the tree above her eyes:
[tex]\[
\text{Total height of the tree} = 5.5 + 49.5 = 55 \text{ feet}
\][/tex]
6. Convert Total Height to Inches:
- To express the total height in inches, multiply the height in feet by 12 (since 1 foot = 12 inches):
[tex]\[
55 \times 12 = 660 \text{ inches}
\][/tex]
Thus, the height of the tree is approximately 55 feet or 660 inches.
1. Understand the Problem:
- Shaina is 5 feet, 6 inches tall.
- She stands 20 feet away from the tree.
- She looks up and sees a bird at an angle of elevation of 68 degrees. We are asked to find the total height of the tree.
2. Convert Units:
- First, convert Shaina's height from feet and inches to only feet. Since she is 5 feet and 6 inches tall, convert inches into feet:
[tex]\[
6 \text{ inches} = \frac{6}{12} \text{ feet} = 0.5 \text{ feet}
\][/tex]
So, Shaina’s height is:
[tex]\[
5 + 0.5 = 5.5 \text{ feet}
\][/tex]
3. Use Trigonometric Ratios:
- The angle of elevation is formed between a horizontal line from Shaina's eye to the base of the tree and her line of sight to the bird.
- Use the tangent of the angle of elevation to find the height of the tree above Shaina’s eyes. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle.
[tex]\[
\tan(68^\circ) = \frac{\text{Height of tree above Shaina's eye}}{20}
\][/tex]
4. Solve for the Height of the Tree above Shaina's Eyes:
- Rearrange the formula to find the height above her eyes:
[tex]\[
\text{Height above eyes} = 20 \times \tan(68^\circ)
\][/tex]
- This height is approximately 49.5 feet.
5. Calculate the Total Height of the Tree:
- To find the total height of the tree, add Shaina's height to the height of the tree above her eyes:
[tex]\[
\text{Total height of the tree} = 5.5 + 49.5 = 55 \text{ feet}
\][/tex]
6. Convert Total Height to Inches:
- To express the total height in inches, multiply the height in feet by 12 (since 1 foot = 12 inches):
[tex]\[
55 \times 12 = 660 \text{ inches}
\][/tex]
Thus, the height of the tree is approximately 55 feet or 660 inches.
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