A boat leaves a port traveling due east at 12 mi/hr. At the same time, another boat leaves the same port traveling northeast at 15 mi/hr. The angle of the
A boat leaves a port traveling due east at 12 mi/hr. At the same time, another boat leaves the same port traveling northeast at 15 mi/hr. The angle of the
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter8: Introduction To Functions
Section8.2: Points, Lines, And Their Graphs
Problem 16WE
Related questions
Question
![### Problem Statement
A boat leaves a port traveling due east at 12 mi/hr. At the same time, another boat leaves the same port traveling northeast at 15 mi/hr. The angle \( \theta \) of the line between the boats is measured relative to due north (see the figure to the right). What is the rate of change of this angle 30 min after the boats leave the port?
### Diagram Explanation
The diagram shows:
- A point labeled "Port" where both boats start.
- One boat traveling east at 12 mi/hr, represented by a horizontal arrow pointing to the right.
- Another boat traveling northeast at 15 mi/hr, represented by an arrow pointing up and to the right at a 45-degree angle.
- The angle \( \theta \) measured between a line extending due north from the port and the northeast-bound boat.
### Mathematical Formulation
**Variables:**
- Let \( x \) be the distance from the port to the eastbound boat.
- Let \( z \) be the distance from the port to the northeast bound boat.
- \( \theta \) is the angle in question.
**Equation Relating \( x \), \( z \), and \( \theta \):**
\[ x = -\frac{\sqrt{2}}{2} z + \frac{\sqrt{2}}{2} z \tan \theta \]
(Type an exact answer. Type any angle measures in radians.)
### Required Rate of Change
Calculate the rate of change of the angle \( \theta \) 30 minutes after the boats leave the port.
**30 minutes (0.5 hours) after departure:**
- Distance traveled east: \( 12 \text{ mi/hr} \times 0.5 \text{ hr} = 6 \text{ miles} \)
- Distance traveled northeast: \( 15 \text{ mi/hr} \times 0.5 \text{ hr} = 7.5 \text{ miles} \)
**Rate of change of the angle \( \theta \):**
\[
\frac{d\theta}{dt} \text{ (rate of change of this angle 30 min after the boats leave the port)}
\]
(Type an exact or rounded answer.)
**The rate of change of \( \theta \) 30 min after the boats leave the port is:**
\[ \boxed{ \quad \} \]
(Round to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa589a0b1-5940-47b0-b798-8fa24b0ff23c%2F8bfc841a-101c-4cd7-84d1-2b55dc0aba91%2Fseb1xau_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
A boat leaves a port traveling due east at 12 mi/hr. At the same time, another boat leaves the same port traveling northeast at 15 mi/hr. The angle \( \theta \) of the line between the boats is measured relative to due north (see the figure to the right). What is the rate of change of this angle 30 min after the boats leave the port?
### Diagram Explanation
The diagram shows:
- A point labeled "Port" where both boats start.
- One boat traveling east at 12 mi/hr, represented by a horizontal arrow pointing to the right.
- Another boat traveling northeast at 15 mi/hr, represented by an arrow pointing up and to the right at a 45-degree angle.
- The angle \( \theta \) measured between a line extending due north from the port and the northeast-bound boat.
### Mathematical Formulation
**Variables:**
- Let \( x \) be the distance from the port to the eastbound boat.
- Let \( z \) be the distance from the port to the northeast bound boat.
- \( \theta \) is the angle in question.
**Equation Relating \( x \), \( z \), and \( \theta \):**
\[ x = -\frac{\sqrt{2}}{2} z + \frac{\sqrt{2}}{2} z \tan \theta \]
(Type an exact answer. Type any angle measures in radians.)
### Required Rate of Change
Calculate the rate of change of the angle \( \theta \) 30 minutes after the boats leave the port.
**30 minutes (0.5 hours) after departure:**
- Distance traveled east: \( 12 \text{ mi/hr} \times 0.5 \text{ hr} = 6 \text{ miles} \)
- Distance traveled northeast: \( 15 \text{ mi/hr} \times 0.5 \text{ hr} = 7.5 \text{ miles} \)
**Rate of change of the angle \( \theta \):**
\[
\frac{d\theta}{dt} \text{ (rate of change of this angle 30 min after the boats leave the port)}
\]
(Type an exact or rounded answer.)
**The rate of change of \( \theta \) 30 min after the boats leave the port is:**
\[ \boxed{ \quad \} \]
(Round to
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 3 steps with 3 images
Recommended textbooks for you
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Holt Mcdougal Larson Pre-algebra: Student Edition…
Algebra
ISBN:
9780547587776
Author:
HOLT MCDOUGAL
Publisher:
HOLT MCDOUGAL
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning