Results 1 to 4 of 4
Like Tree2Thanks
  • 2 Post By HallsofIvy

Math Help - Angular Rate of Change

  1. #1
    Banned
    Joined
    Jan 2013
    From
    New York City
    Posts
    658
    Thanks
    7

    Angular Rate of Change

    An plane flies at a height of 5 miles to a point directly over an observer. Consider theta and x. Let x be the distance from the observer to the point directly below the plane. I made a picture of a right triangle.

    The question tells me to write theta as a function of x. So, my answer is theta = arctan(5miles/x). Is this correct?

    Part B: The speed of the plane is 400 mph. Find d(theta)/dt when x is 10 miles and when x is 3 miles.

    The answer must be in radians because I am searching for angle theta. Is this right?

    I found d(theta)/dt to be -5/x^2 divided by 1 + (25/x^2). Is this the correct function? It is a complex fraction.

    Afterward, I simply plugged x = 10 and x = 3 individually and simplified.

    For x = 10, my answer is -1/25.
    For x = 3, my answer is -5/34.

    Is this right?

    This question is from my single variable calculus text, chapter on finding the derivative of inverse functions. Thanks....
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,780
    Thanks
    1521

    Re: Angular Rate of Change

    Quote Originally Posted by nycmath View Post
    An plane flies at a height of 5 miles to a point directly over an observer. Consider theta and x. Let x be the distance from the observer to the point directly below the plane. I made a picture of a right triangle.

    The question tells me to write theta as a function of x. So, my answer is theta = arctan(5miles/x). Is this correct?
    Yes, it is. Note that theta= arccot(x/5) is also correct.

    Part B: The speed of the plane is 400 mph. Find d(theta)/dt when x is 10 miles and when x is 3 miles.

    The answer must be in radians because I am searching for angle theta. Is this right?
    No, the answer must be in radians per hour (or in terms of an angle measure over a time measure) because you area asked for the rate of change of theta.

    I found d(theta)/dt to be -5/x^2 divided by 1 + (25/x^2). Is this the correct function? It is a complex fraction.
    No, that is not correct. You have missed one important point.

    You have correctly written theta= arctan(5/x). The derivative of arctan(u) is \dfrac{1}{1+u^2}\dfrac{du}{dx}. Here, u= 5/x= 5x^{-1} so du/dx= -5x^{-2}= -5/x^2 so, the derivative of arctan(5/x) with respect to x is \dfrac{-5/x^2}{1+ 25/x^2}. It might be better to multiply both numerator and denominator by x^2 to get \dfrac{-5}{x^2+ 25}.

    However, the crucial point is that d\theta/dt, which is what was asked, is \dfrac{-5}{x^2+ 25}\dfrac{dx}{dt}.

    Afterward, I simply plugged x = 10 and x = 3 individually and simplified.

    For x = 10, my answer is -1/25.
    For x = 3, my answer is -5/34.
    If x= 10, x^2+ 25= 125 so \dfrac{-5}{x^2+ 1}= \dfrac{-5}{125}= -\dfrac{1}{25}. However, that is d\theta/dx, not d\theta/dt! You have not used the fact that dx/dt= 400 mph.

    Is this right?
    No. It isn't. You did all of the calculations correctly but you did not answer the question that was asked!

    This question is from my single variable calculus text, chapter on finding the derivative of inverse functions. Thanks....
    Last edited by HallsofIvy; November 7th 2013 at 08:57 AM.
    Thanks from topsquark and nycmath
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Jan 2013
    From
    New York City
    Posts
    658
    Thanks
    7

    Re: Angular Rate of Change

    I truly appreciate your help and guidance with this question that is not explained in my text.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Nov 2013
    From
    Bestwebuys
    Posts
    6

    Re: Angular Rate of Change

    Bi viết hay!

    Ruột bt Montblanc chnh hng tại Việt Nam - 400 Trần Hưng Đạo, P.2, Q.5, Tp.Hồ Ch Minh Thứ 2 - 7 : 8:00am - 5:30pm Cc sản phẩm ruột bt Montblanc bao gồm: ballpoint, fineliner, mozart v rollerball.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. rate of change
    Posted in the Calculus Forum
    Replies: 6
    Last Post: June 4th 2011, 10:18 PM
  2. Replies: 3
    Last Post: April 12th 2011, 09:51 AM
  3. angular rate of change
    Posted in the Calculus Forum
    Replies: 1
    Last Post: January 19th 2010, 04:03 PM
  4. rate of change help
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 5th 2009, 07:13 PM
  5. rate of change help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: August 7th 2008, 09:22 AM

Search Tags


/mathhelpforum @mathhelpforum