Approximation and error in functions of multiple variables - help needed

In summary: The baseball player is playing approximately 310 feet from a TV camera that is behind home plate. A batter hits a fly ball that goes to a wall 400 feet from the camera.The camera turns 12 degrees to follow the play. Approximate the number of feet the fielder has to make the catch. The maximum possible error in measuring the rotation of the camera is 1 degree. If the position of the fielder could be in error by as much as 5 feet, and the maximum error in measuring the rotation of the camera is 1 degree, approximate the maximum possible error in your answer from part (a).
  • #1
Vampire
11
0

Homework Statement


A baseball player is playing approximately 310 feet from a TV camera that is behind home plate. A batter hits a fly ball that goes to a wall 400 feet from the camera.
(a) The Camera turns 12 degrees to follow the play. Approximate the number of feet the fielder has to make the catch.
(b) If the position of the fielder could be in error by as much as 5 feet, and the maximum error in measuring the rotation of the camera is 1 degree, approximate the maximum possible error in your answer from part (a).

Verbatim.

Homework Equations


The law of cosines.
c2 = a2 + b2 - 2abcos([tex]\theta[/tex])
Linear approximation at (a,b,c):
L(x,y,z) = f(a,b,c) + fx(x,y,z)(x-a) + fy(x,y,z)(y-b) + fz(x,y,z)(z-c)


The Attempt at a Solution



I used the law of cosines to set up a function of 3 variables, and I left one variable constant.

c = f(a,b,[tex]\theta[/tex]) = the distance that the fielder moves.
a = the initial distance of the fielder from the camera.
b = the position of the ball after the play. This was constant at 400.
[tex]\theta[/tex] = the angle the camera moves in radians.

f(a,b,[tex]\theta[/tex]) = ( a2 + 4002 - 800*a*cos([tex]\theta[/tex]) )1/2

After that I used a linear approximation to approximate the distance the player must move (since the problem instructs me to do so). I got an answer nowhere near the exact value I calculated.

I do not know how to find the maximum possible error in (b).

I have the feeling that I did not set this problem up correctly to begin with. Did I?

EDIT: I got this feeling due to the fact that f(a,b,c) = 0 in my setup

fa = ( a - 400cos([tex]\theta[/tex]) ) / ( a2 +4002 -a*800cos([tex]\theta[/tex]) )1/2
f[tex]\theta[/tex] = ( 400asin([tex]\theta[/tex]) ) / ( a2 +4002 -a*800cos([tex]\theta[/tex]) )1/2
 
Last edited:
Physics news on Phys.org
  • #2
I would NOT interpret "approximate the number of feet" as meaning to use a linear approximation. You will be approximating by simply calculating f(a,b,[itex]\theta[/itex]) since the value of cos(12) that your calculator gives you is only approximate.

For (b) you can use the approximation [itex]df= f_a da+ f_b db+ f_\theta d\theta[/itex], with da= db= 5 and [itex]d\theta= 1[/itex] degree (so you will need to convert this to radians).
 
  • #3
Thank you. That clears things up.
 

What is approximation and error in functions of multiple variables?

Approximation and error in functions of multiple variables refers to the process of estimating the value of a function with more than one input variable and the difference between the estimated value and the actual value.

Why is approximation and error important in scientific research?

Approximation and error are important in scientific research because they allow scientists to make predictions and draw conclusions based on limited data. It also helps to identify areas of uncertainty and improve the accuracy of experiments and models.

What are some common methods for approximating functions of multiple variables?

Some common methods for approximating functions of multiple variables include Taylor series, linear approximation, and interpolation methods such as regression and spline interpolation.

What are the sources of error in approximating functions of multiple variables?

The sources of error in approximating functions of multiple variables can include rounding errors, truncation errors, and errors from using simplified models or assumptions. Measurement errors and limitations of the instruments used can also contribute to the error.

How can we minimize error in approximating functions of multiple variables?

To minimize error in approximating functions of multiple variables, we can use more precise measurement tools, increase the number of data points, and use more accurate and complex models. It is also important to understand the sources of error and try to reduce or eliminate them as much as possible.

Similar threads

  • Calculus and Beyond Homework Help
Replies
34
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus and Beyond Homework Help
Replies
9
Views
462
  • Calculus and Beyond Homework Help
Replies
1
Views
633
  • Calculus and Beyond Homework Help
Replies
8
Views
348
  • Calculus and Beyond Homework Help
Replies
3
Views
742
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
11
Views
947
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Back
Top