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
Trigonometry (MindTap Course List)
8th Edition
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Charles P. McKeague, Mark D. Turner
Chapter3: Radian Measure
Section3.5: Velocities
Problem 54PS: Velocity of a Ferris Wheel Use Figure 7 as a model of the Ferris wheel called Colossus that is built...
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![### 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
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