16. Julia is training to track airplanes with a radar. Her system detects an aircraft flying at a constant speed heading in a straight line passing directly over her location. She sees the aircraft at an angle of elevation of 15° and observes that it maintains a constant altitude of 6,250 feet. One minute later, she sees the airplane at an angle of elevation of 52°. A. How far has the airplane traveled, to the nearest foot? B. Determine and state the speed of the aircraft, to the nearest mile per hour. [Draw a diagram and show all work]
16. Julia is training to track airplanes with a radar. Her system detects an aircraft flying at a constant speed heading in a straight line passing directly over her location. She sees the aircraft at an angle of elevation of 15° and observes that it maintains a constant altitude of 6,250 feet. One minute later, she sees the airplane at an angle of elevation of 52°. A. How far has the airplane traveled, to the nearest foot? B. Determine and state the speed of the aircraft, to the nearest mile per hour. [Draw a diagram and show all work]
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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This is a geometry question.
![### Problem Statement
Julia is training to track airplanes with a radar. Her system detects an aircraft flying at a constant speed heading in a straight line passing directly over her location. She sees the aircraft at an angle of elevation of 15° and observes that it maintains a constant altitude of 6,250 feet. One minute later, she sees the airplane at an angle of elevation of 52°.
**A. How far has the airplane traveled, to the nearest foot?**
**B. Determine and state the speed of the aircraft, to the nearest mile per hour. [Draw a diagram and show all work]**
### Detailed Explanation
**A. Calculating Distance Traveled**
To determine how far the airplane has traveled, use trigonometry. The initial and final angles of elevation and the known altitude will allow you to calculate the horizontal distance traveled.
Let's denote:
- \( h = 6,250 \text{ feet} \)
- \( \theta_1 = 15^\circ \)
- \( \theta_2 = 52^\circ \)
First, calculate the distances from Julia to the airplane at each observed angle of elevation:
- Distance for \( \theta_1 \): \( d_1 \)
- Distance for \( \theta_2 \): \( d_2 \)
Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan(15^\circ) = \frac{6,250}{d_1} \]
\[ \tan(52^\circ) = \frac{6,250}{d_2} \]
Solving these equations for \( d_1 \) and \( d_2 \):
\[ d_1 = \frac{6,250}{\tan(15^\circ)} \]
\[ d_2 = \frac{6,250}{\tan(52^\circ)} \]
Insert the values of tangent for \( 15^\circ \) and \( 52^\circ \) (which can be found using a calculator):
\[ d_1 \approx \frac{6,250}{0.2679} \approx 23,327 \text{ feet} \]
\[ d_2 \approx \frac{6,250}{1.2799} \approx 4,882 \text{ feet} \]
The horizontal distance traveled by the airplane is:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc52b8848-81e8-44e6-b993-71dacd213084%2F620c1e5f-d239-44b0-abdb-b38e0995cf9a%2Fyexq8wr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Julia is training to track airplanes with a radar. Her system detects an aircraft flying at a constant speed heading in a straight line passing directly over her location. She sees the aircraft at an angle of elevation of 15° and observes that it maintains a constant altitude of 6,250 feet. One minute later, she sees the airplane at an angle of elevation of 52°.
**A. How far has the airplane traveled, to the nearest foot?**
**B. Determine and state the speed of the aircraft, to the nearest mile per hour. [Draw a diagram and show all work]**
### Detailed Explanation
**A. Calculating Distance Traveled**
To determine how far the airplane has traveled, use trigonometry. The initial and final angles of elevation and the known altitude will allow you to calculate the horizontal distance traveled.
Let's denote:
- \( h = 6,250 \text{ feet} \)
- \( \theta_1 = 15^\circ \)
- \( \theta_2 = 52^\circ \)
First, calculate the distances from Julia to the airplane at each observed angle of elevation:
- Distance for \( \theta_1 \): \( d_1 \)
- Distance for \( \theta_2 \): \( d_2 \)
Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]
\[ \tan(15^\circ) = \frac{6,250}{d_1} \]
\[ \tan(52^\circ) = \frac{6,250}{d_2} \]
Solving these equations for \( d_1 \) and \( d_2 \):
\[ d_1 = \frac{6,250}{\tan(15^\circ)} \]
\[ d_2 = \frac{6,250}{\tan(52^\circ)} \]
Insert the values of tangent for \( 15^\circ \) and \( 52^\circ \) (which can be found using a calculator):
\[ d_1 \approx \frac{6,250}{0.2679} \approx 23,327 \text{ feet} \]
\[ d_2 \approx \frac{6,250}{1.2799} \approx 4,882 \text{ feet} \]
The horizontal distance traveled by the airplane is:
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