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Linear approximation and errors

  1. Oct 23, 2007 #1
    [SOLVED] Linear approximation

    1. The problem statement, all variables and given/known data
    Juan measures the circumference C of a spherical ball at 40cm and computes the ball's volume V. Estimate the maximum possible error in V if the error in C is as most 2cm. Recall that C=2(pi)r and V=(4/3)pi(r)


    2. Relevant equations
    deltaf - f'(a)h, a= 40, V=(4/3)pi(C/2pi)^3


    3. The attempt at a solution

    im not sure exactly how to compute to find the final answer but i believe that a = 40cm + or - 2cm and that somehow solving for r using circumference and then plugging it into the volume equation. if you can point me in the right direction i would appreciate it, thanks.
     
  2. jcsd
  3. Oct 23, 2007 #2

    EnumaElish

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    The easiest way is to assume maximum error (plus, then minus). Then take the difference between the calculated volume and true volume.

    Or you could try expressing V as a function of C.
     
  4. Oct 23, 2007 #3
    thanks, that actually helps me a lot.
     
  5. Oct 23, 2007 #4
    after solving for the volume when the radius is (21/pi) which is a +2cm error i got the answer 1251.1cm^3. when i solved for the volume when the radius was (19/pi) a -2cm error i got 926.6cm^3. last i solved for the actual measured raius which was 40 and found the radius to be (20/pi) when i solved for this i got 1080.8. i then subtracted 926.6 (-2cm error) from this answer and got 154.2. when i subtracted 1080.8 from 1251.1 (+2cm error minus actual) i got the answer 170.3. I'm guessing this means that the maximum error is 170.3cm^3. can anyone verify this?
     
  6. Oct 24, 2007 #5

    HallsofIvy

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    V=(4/3)pi(C/2pi)^3= C^3/(6pi^2) is exactly what you want. What is the derivative of V with respect to C?
     
  7. Oct 24, 2007 #6
    4pi(c/2pi)^2
     
  8. Oct 24, 2007 #7
    i think i finally got the answer to be 324.22pi by plugging in 40 to the above equation for C and then multipliing by delta x which in this case is 2
     
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