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.
Question
![**Problem 42: Calculating the Height of a Drive-In Movie Screen**
In this problem, you are positioned 50 feet away from the screen at a drive-in movie theater. The angle of elevation from your position to the top of the screen is 60 degrees. The task is to determine the height of the movie screen.
**Diagram Explanation:**
Although there is no diagram provided, you can visualize this scenario with a right triangle. The base of the triangle represents the distance between you and the screen (50 feet), the height of the triangle represents the height of the screen that needs to be calculated, and the hypotenuse is the line of sight from your position to the top of the screen. The angle of elevation (60 degrees) is formed between the base and the hypotenuse.
**Solution:**
To solve this problem, you can use the tangent trigonometric function, which relates the angle of elevation to the opposite and adjacent sides of the right triangle.
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case, the angle is 60 degrees, the opposite side is the height of the screen (\(h\)), and the adjacent side is the distance to the screen (50 feet).
\[ \tan(60^\circ) = \frac{h}{50} \]
Since \( \tan(60^\circ) = \sqrt{3} \approx 1.732 \):
\[ 1.732 = \frac{h}{50} \]
To find \( h \), multiply both sides by 50:
\[ h = 1.732 \times 50 \]
\[ h \approx 86.6 \]
So, the height of the screen is approximately 86.6 feet.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb349b7fd-7f7c-4cb3-91c7-7dc92458a5c7%2Fc4a3440e-1ee1-4e0f-918c-97982bd80579%2Ft8q7oj6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 42: Calculating the Height of a Drive-In Movie Screen**
In this problem, you are positioned 50 feet away from the screen at a drive-in movie theater. The angle of elevation from your position to the top of the screen is 60 degrees. The task is to determine the height of the movie screen.
**Diagram Explanation:**
Although there is no diagram provided, you can visualize this scenario with a right triangle. The base of the triangle represents the distance between you and the screen (50 feet), the height of the triangle represents the height of the screen that needs to be calculated, and the hypotenuse is the line of sight from your position to the top of the screen. The angle of elevation (60 degrees) is formed between the base and the hypotenuse.
**Solution:**
To solve this problem, you can use the tangent trigonometric function, which relates the angle of elevation to the opposite and adjacent sides of the right triangle.
\[ \tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} \]
In this case, the angle is 60 degrees, the opposite side is the height of the screen (\(h\)), and the adjacent side is the distance to the screen (50 feet).
\[ \tan(60^\circ) = \frac{h}{50} \]
Since \( \tan(60^\circ) = \sqrt{3} \approx 1.732 \):
\[ 1.732 = \frac{h}{50} \]
To find \( h \), multiply both sides by 50:
\[ h = 1.732 \times 50 \]
\[ h \approx 86.6 \]
So, the height of the screen is approximately 86.6 feet.
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