Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 72RE
Related questions
Question
![**Problem Description:**
A paratrooper is supposed to jump out of a helicopter and make his way to supplies that are marked by a black dot in the diagram below. The angle of elevation from the supplies to the helicopter is 37 degrees, and the distance from the supplies directly to the helicopter (the line of sight) is 2.1 kilometers. Find the distance from the point where the paratrooper will land on the ground to the supplies. Round your answer to the nearest tenth.
**Diagram:**
The diagram includes the following elements:
- A helicopter positioned in the air.
- A paratrooper jumping out of the helicopter.
- A straight dashed line (vertical) from the helicopter, representing the height of the helicopter from the ground.
- A black dot on the ground marking the location of the supplies.
- A diagonal dotted line from the helicopter to the black dot, labeled "2.1 km."
- An angle of elevation, marked as 37°, between the line of sight and the ground.
**Solution:**
To find the distance from the point where the paratrooper lands on the ground to the supplies, we will use trigonometric functions.
Using the cosine function for the given right-angled triangle:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Given:
\[ \theta = 37^\circ \]
\[ \text{Hypotenuse} = 2.1 \text{ km} \]
Let's denote the distance from the ground point directly below the helicopter to the supplies as \( d \).
\[ \cos(37^\circ) = \frac{d}{2.1} \]
To solve for \( d \):
\[ d = 2.1 \times \cos(37^\circ) \]
Using the approximation \(\cos(37^\circ) \approx 0.7986\):
\[ d \approx 2.1 \times 0.7986 \]
\[ d \approx 1.67706 \text{ km} \]
Rounding to the nearest tenth:
\[ d \approx 1.7 \text{ km} \]
**Answer Box:**
The distance from the paratrooper's point of landing to the supplies is
\[ \boxed{1.7} \] kilometers. Round your answer to the nearest tenth.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9bb604e-ff67-4579-a06e-ac819480205b%2F635ae7d0-b244-4679-93f4-a255da256259%2Fpz9zzed_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
A paratrooper is supposed to jump out of a helicopter and make his way to supplies that are marked by a black dot in the diagram below. The angle of elevation from the supplies to the helicopter is 37 degrees, and the distance from the supplies directly to the helicopter (the line of sight) is 2.1 kilometers. Find the distance from the point where the paratrooper will land on the ground to the supplies. Round your answer to the nearest tenth.
**Diagram:**
The diagram includes the following elements:
- A helicopter positioned in the air.
- A paratrooper jumping out of the helicopter.
- A straight dashed line (vertical) from the helicopter, representing the height of the helicopter from the ground.
- A black dot on the ground marking the location of the supplies.
- A diagonal dotted line from the helicopter to the black dot, labeled "2.1 km."
- An angle of elevation, marked as 37°, between the line of sight and the ground.
**Solution:**
To find the distance from the point where the paratrooper lands on the ground to the supplies, we will use trigonometric functions.
Using the cosine function for the given right-angled triangle:
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Given:
\[ \theta = 37^\circ \]
\[ \text{Hypotenuse} = 2.1 \text{ km} \]
Let's denote the distance from the ground point directly below the helicopter to the supplies as \( d \).
\[ \cos(37^\circ) = \frac{d}{2.1} \]
To solve for \( d \):
\[ d = 2.1 \times \cos(37^\circ) \]
Using the approximation \(\cos(37^\circ) \approx 0.7986\):
\[ d \approx 2.1 \times 0.7986 \]
\[ d \approx 1.67706 \text{ km} \]
Rounding to the nearest tenth:
\[ d \approx 1.7 \text{ km} \]
**Answer Box:**
The distance from the paratrooper's point of landing to the supplies is
\[ \boxed{1.7} \] kilometers. Round your answer to the nearest tenth.
![### Angle Calculation in a Right Triangle
**Problem Statement:**
Find the angle measure represented by \( x \).
The diagram provided is of a right triangle with the following side measurements:
- Opposite side to angle \( x \): 7 units
- Adjacent side to angle \( x \): 14 units
**Diagram:**
A right triangle is shown, where:
- One angle is marked as \( x \).
- The side opposite to \( x \) is labeled as 7 units.
- The side adjacent to \( x \) is labeled as 14 units.
**Task:**
Determine the angle measure \( x \) in degrees.
**Solution:**
To find the angle \( x \), we can use the tangent function in trigonometry, which is defined as:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, the opposite side is 7 units, and the adjacent side is 14 units.
\[
\tan(x) = \frac{7}{14} = \frac{1}{2}
\]
So,
\[
x = \tan^{-1}\left(\frac{1}{2}\right)
\]
Use a calculator to find the inverse tangent (arctan) of \( \frac{1}{2} \).
---
**Interactive Component:**
The angle measure represented by \( x \) is: [type your answer...] degrees](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9bb604e-ff67-4579-a06e-ac819480205b%2F635ae7d0-b244-4679-93f4-a255da256259%2F7lpqf0v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Angle Calculation in a Right Triangle
**Problem Statement:**
Find the angle measure represented by \( x \).
The diagram provided is of a right triangle with the following side measurements:
- Opposite side to angle \( x \): 7 units
- Adjacent side to angle \( x \): 14 units
**Diagram:**
A right triangle is shown, where:
- One angle is marked as \( x \).
- The side opposite to \( x \) is labeled as 7 units.
- The side adjacent to \( x \) is labeled as 14 units.
**Task:**
Determine the angle measure \( x \) in degrees.
**Solution:**
To find the angle \( x \), we can use the tangent function in trigonometry, which is defined as:
\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}}
\]
Here, the opposite side is 7 units, and the adjacent side is 14 units.
\[
\tan(x) = \frac{7}{14} = \frac{1}{2}
\]
So,
\[
x = \tan^{-1}\left(\frac{1}{2}\right)
\]
Use a calculator to find the inverse tangent (arctan) of \( \frac{1}{2} \).
---
**Interactive Component:**
The angle measure represented by \( x \) is: [type your answer...] degrees
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, geometry and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
