The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70°, how tall is the Space Needle? Round to the nearest meter.
The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70°, how tall is the Space Needle? Round to the nearest meter.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
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![### Calculating the Height of the Seattle Space Needle
The problem presented involves determining the height of the Seattle Space Needle using trigonometry. Here's the given information and the steps to solve it:
**Problem Statement:**
"The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70 degrees, how tall is the Space Needle? Round to the nearest meter."
**Solution Explanation:**
To solve this problem, you can use the tangent function in trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
1. **Identify the variables:**
- Let \( h \) represent the height of the Space Needle.
- The shadow length is the adjacent side, which is 67 meters.
- The angle of elevation is given as 70 degrees.
2. **Set up the tangent function:**
- The tangent of the angle of elevation (70 degrees) is equal to the height of the Space Needle divided by the length of its shadow:
\[
\tan(70^\circ) = \frac{h}{67}
\]
3. **Solve for \( h \):**
- Rearrange the equation to solve for \( h \):
\[
h = 67 \times \tan(70^\circ)
\]
4. **Calculate the value:**
- Use a calculator to find the tangent of 70 degrees and multiply by 67 meters:
\[
\tan(70^\circ) \approx 2.747
\]
\[
h = 67 \times 2.747 \approx 184.05
\]
5. **Round to the nearest meter:**
\[
h \approx 184 \text{ meters}
\]
**Conclusion:**
The height of the Seattle Space Needle is approximately 184 meters.
This calculation demonstrates how trigonometry can be used to determine heights and distances that are otherwise difficult to measure directly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa289933c-d83f-4ce1-b14d-e69289a7491a%2Fa8571708-cb1e-4483-98da-6c93523fc086%2Fq7w1sds_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Height of the Seattle Space Needle
The problem presented involves determining the height of the Seattle Space Needle using trigonometry. Here's the given information and the steps to solve it:
**Problem Statement:**
"The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70 degrees, how tall is the Space Needle? Round to the nearest meter."
**Solution Explanation:**
To solve this problem, you can use the tangent function in trigonometry. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side.
1. **Identify the variables:**
- Let \( h \) represent the height of the Space Needle.
- The shadow length is the adjacent side, which is 67 meters.
- The angle of elevation is given as 70 degrees.
2. **Set up the tangent function:**
- The tangent of the angle of elevation (70 degrees) is equal to the height of the Space Needle divided by the length of its shadow:
\[
\tan(70^\circ) = \frac{h}{67}
\]
3. **Solve for \( h \):**
- Rearrange the equation to solve for \( h \):
\[
h = 67 \times \tan(70^\circ)
\]
4. **Calculate the value:**
- Use a calculator to find the tangent of 70 degrees and multiply by 67 meters:
\[
\tan(70^\circ) \approx 2.747
\]
\[
h = 67 \times 2.747 \approx 184.05
\]
5. **Round to the nearest meter:**
\[
h \approx 184 \text{ meters}
\]
**Conclusion:**
The height of the Seattle Space Needle is approximately 184 meters.
This calculation demonstrates how trigonometry can be used to determine heights and distances that are otherwise difficult to measure directly.
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