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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![**Title: Solving for the Height of a Tree Using Trigonometry**
**Problem:**
A tree casts a shadow 76 feet long. The angle of elevation of the sun is 49 degrees. Find the height of the tree.
**Illustration and Explanation:**
The image shows a right triangle with the following details:
- The hypotenuse is the line of sight from the top of the tree to the tip of the shadow.
- The base of the triangle (adjacent side) represents the shadow of the tree, which is 76 feet long.
- The angle of elevation (the angle formed between the base and the hypotenuse) is given as 49 degrees.
- The height of the tree represents the opposite side of the triangle, which we need to find.
**Trigonometric Solution:**
1. **Identify the trigonometric function to use:**
To find the height of the tree, we use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (length of the shadow).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
2. **Substitute the known values into the equation:**
Here, \(\theta = 49^\circ\) and the length of the shadow is 76 feet.
\[
\tan(49^\circ) = \frac{\text{height}}{76 \text{ ft}}
\]
3. **Solve for the height:**
Rearrange the equation to solve for the height of the tree:
\[
\text{height} = 76 \text{ ft} \times \tan(49^\circ)
\]
4. **Calculate the height:**
Use a calculator to find the tangent of 49 degrees:
\[
\tan(49^\circ) \approx 1.1504
\]
\[
\text{height} \approx 76 \text{ ft} \times 1.1504 \approx 87.43 \text{ ft}
\]
Therefore, the height of the tree is approximately **87.43 feet**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc476fe49-bd29-4169-8079-6a3850fe1835%2Fbf5dc88f-5cc8-4028-86ab-b1880861abbc%2Fu3lug5p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Solving for the Height of a Tree Using Trigonometry**
**Problem:**
A tree casts a shadow 76 feet long. The angle of elevation of the sun is 49 degrees. Find the height of the tree.
**Illustration and Explanation:**
The image shows a right triangle with the following details:
- The hypotenuse is the line of sight from the top of the tree to the tip of the shadow.
- The base of the triangle (adjacent side) represents the shadow of the tree, which is 76 feet long.
- The angle of elevation (the angle formed between the base and the hypotenuse) is given as 49 degrees.
- The height of the tree represents the opposite side of the triangle, which we need to find.
**Trigonometric Solution:**
1. **Identify the trigonometric function to use:**
To find the height of the tree, we use the tangent function, which relates the angle of elevation to the opposite side (height of the tree) and the adjacent side (length of the shadow).
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
2. **Substitute the known values into the equation:**
Here, \(\theta = 49^\circ\) and the length of the shadow is 76 feet.
\[
\tan(49^\circ) = \frac{\text{height}}{76 \text{ ft}}
\]
3. **Solve for the height:**
Rearrange the equation to solve for the height of the tree:
\[
\text{height} = 76 \text{ ft} \times \tan(49^\circ)
\]
4. **Calculate the height:**
Use a calculator to find the tangent of 49 degrees:
\[
\tan(49^\circ) \approx 1.1504
\]
\[
\text{height} \approx 76 \text{ ft} \times 1.1504 \approx 87.43 \text{ ft}
\]
Therefore, the height of the tree is approximately **87.43 feet**.
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