A golfer makes a successful chip shot to the green. Suppose the path of the ball from the moment it s struck to the moment it hits the green S described by y=13.51x-0.62x² where x E [0, b where x is the horizontal distance (in yards) From the point where the ball was struck, nd y is the vertical distance (in yards) abov he fairway. a.) set up the integral (to just before integrat to find the distance the ball trave (the arc length) from the moment it Struck to the moment it hits green. b.) If the ball travels 42 yards Chori down the fairway, find the arc le the ball travels. Round the near hundred.
A golfer makes a successful chip shot to the green. Suppose the path of the ball from the moment it s struck to the moment it hits the green S described by y=13.51x-0.62x² where x E [0, b where x is the horizontal distance (in yards) From the point where the ball was struck, nd y is the vertical distance (in yards) abov he fairway. a.) set up the integral (to just before integrat to find the distance the ball trave (the arc length) from the moment it Struck to the moment it hits green. b.) If the ball travels 42 yards Chori down the fairway, find the arc le the ball travels. Round the near hundred.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Title: Calculating the Arc Length of a Golf Ball's Path**
A golfer makes a successful chip shot to the green. Suppose the path of the ball from the moment it is struck to the moment it hits the green is described by the equation:
\[ y = 13.51x - 0.62x^2 \]
where \( x \) is the horizontal distance (in yards) from the point where the ball was struck, and \( y \) is the vertical distance (in yards) above the fairway.
Given these parameters:
- \( x \) belongs to the interval \([0, b]\)
### Problem Statement:
#### Part (a):
Set up the integral (to just before integrating) to find the distance the ball travels (the arc length) from the moment it is struck to the moment it hits the green.
#### Part (b):
If the ball travels 42 yards (horizontally) down the fairway, find the arc length the ball travels. Round to the nearest hundred.
### Solution Outline:
#### Part (a):
To find the arc length \( L \) of the path of the ball, we use the formula for the arc length of a curve given by the function \( y = f(x) \):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \]
First, we need to find \( \frac{dy}{dx} \):
\[ y = 13.51x - 0.62x^2 \]
\[ \frac{dy}{dx} = 13.51 - 1.24x \]
Thus, the integrand becomes:
\[ 1 + \left( 13.51 - 1.24x \right)^2 \]
Therefore, the integral to find the arc length \( L \) is:
\[ L = \int_{0}^{b} \sqrt{1 + \left(13.51 - 1.24x \right)^2 } \, dx \]
#### Part (b):
If the ball travels 42 yards horizontally (\( x = 42 \)):
\[ L = \int_{0}^{42} \sqrt{1 + \left(13.51 - 1.24x \right)^2 } \, dx \]
To solve this integral, numerical methods or software tools](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a617c73-16c8-405c-b9c4-72dd4741d892%2F7960d971-a901-42a6-88d4-3bd34ce608c7%2Fqctbh2e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating the Arc Length of a Golf Ball's Path**
A golfer makes a successful chip shot to the green. Suppose the path of the ball from the moment it is struck to the moment it hits the green is described by the equation:
\[ y = 13.51x - 0.62x^2 \]
where \( x \) is the horizontal distance (in yards) from the point where the ball was struck, and \( y \) is the vertical distance (in yards) above the fairway.
Given these parameters:
- \( x \) belongs to the interval \([0, b]\)
### Problem Statement:
#### Part (a):
Set up the integral (to just before integrating) to find the distance the ball travels (the arc length) from the moment it is struck to the moment it hits the green.
#### Part (b):
If the ball travels 42 yards (horizontally) down the fairway, find the arc length the ball travels. Round to the nearest hundred.
### Solution Outline:
#### Part (a):
To find the arc length \( L \) of the path of the ball, we use the formula for the arc length of a curve given by the function \( y = f(x) \):
\[ L = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } \, dx \]
First, we need to find \( \frac{dy}{dx} \):
\[ y = 13.51x - 0.62x^2 \]
\[ \frac{dy}{dx} = 13.51 - 1.24x \]
Thus, the integrand becomes:
\[ 1 + \left( 13.51 - 1.24x \right)^2 \]
Therefore, the integral to find the arc length \( L \) is:
\[ L = \int_{0}^{b} \sqrt{1 + \left(13.51 - 1.24x \right)^2 } \, dx \]
#### Part (b):
If the ball travels 42 yards horizontally (\( x = 42 \)):
\[ L = \int_{0}^{42} \sqrt{1 + \left(13.51 - 1.24x \right)^2 } \, dx \]
To solve this integral, numerical methods or software tools
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
