Two sides of a triangle have lengths 8 m and 13 m. The angle between them is increasing at a rate of 0.03radians /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 7/3 radians. Note: The length of the side C of a triangle is given by the law of cosines: c² = a² + b² - 2ab cos(6), where a and b are the other two sides and is the angle between them. Check the Trigonometric formulas at the end of the book on Reference Page 2.
Two sides of a triangle have lengths 8 m and 13 m. The angle between them is increasing at a rate of 0.03radians /min. How fast is the length of the third side increasing when the angle between the sides of fixed length is 7/3 radians. Note: The length of the side C of a triangle is given by the law of cosines: c² = a² + b² - 2ab cos(6), where a and b are the other two sides and is the angle between them. Check the Trigonometric formulas at the end of the book on Reference Page 2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Two sides of a triangle have lengths 8 m and 13 m. The angle between them is increasing at a rate of 0.03 radians/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is \( \pi/3 \) radians?
**Solution:**
To solve this problem, we use the law of cosines which states:
\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \]
where \( a \) and \( b \) are the lengths of the two sides, and \( \theta \) is the angle between them. We are given:
- \( a = 8 \, \text{m} \)
- \( b = 13 \, \text{m} \)
- \(\frac{d\theta}{dt} = 0.03 \, \text{radians/min} \)
- \( \theta = \pi/3 \)
**Steps to Find \(\frac{dc}{dt}\):**
1. Differentiate the law of cosines with respect to time \( t \):
\[
2c \frac{dc}{dt} = 2ab \sin(\theta) \frac{d\theta}{dt}
\]
2. Substitute \( \theta = \pi/3 \), \( a = 8 \), \( b = 13 \), and \( \frac{d\theta}{dt} = 0.03 \).
3. Find \( c \) using the law of cosines:
\[
c^2 = 8^2 + 13^2 - 2(8)(13)\cos(\pi/3)
\]
4. Calculate \( c \) and plug it into the differentiated equation to solve for \( \frac{dc}{dt} \).
This allows us to find how fast the third side is increasing when the angle between the two given sides is \( \pi/3 \) radians. It provides an application of trigonometric formulas and differentiation in the context of a real-world problem.
**Note:** Check the Trigonometric formulas at the end of the book on Reference Page 2 for further details.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F220673f3-4665-41cf-82e7-eaf0663df6ee%2F92255f8a-38db-41a4-84ef-e5b9d2da9d86%2F3u5xtg_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Two sides of a triangle have lengths 8 m and 13 m. The angle between them is increasing at a rate of 0.03 radians/min. How fast is the length of the third side increasing when the angle between the sides of fixed length is \( \pi/3 \) radians?
**Solution:**
To solve this problem, we use the law of cosines which states:
\[ c^2 = a^2 + b^2 - 2ab \cos(\theta) \]
where \( a \) and \( b \) are the lengths of the two sides, and \( \theta \) is the angle between them. We are given:
- \( a = 8 \, \text{m} \)
- \( b = 13 \, \text{m} \)
- \(\frac{d\theta}{dt} = 0.03 \, \text{radians/min} \)
- \( \theta = \pi/3 \)
**Steps to Find \(\frac{dc}{dt}\):**
1. Differentiate the law of cosines with respect to time \( t \):
\[
2c \frac{dc}{dt} = 2ab \sin(\theta) \frac{d\theta}{dt}
\]
2. Substitute \( \theta = \pi/3 \), \( a = 8 \), \( b = 13 \), and \( \frac{d\theta}{dt} = 0.03 \).
3. Find \( c \) using the law of cosines:
\[
c^2 = 8^2 + 13^2 - 2(8)(13)\cos(\pi/3)
\]
4. Calculate \( c \) and plug it into the differentiated equation to solve for \( \frac{dc}{dt} \).
This allows us to find how fast the third side is increasing when the angle between the two given sides is \( \pi/3 \) radians. It provides an application of trigonometric formulas and differentiation in the context of a real-world problem.
**Note:** Check the Trigonometric formulas at the end of the book on Reference Page 2 for further details.
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