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Math Help - Rate of Change with radians

  1. #1
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    Rate of Change with radians

    This confused me a little bit:



    Suppose that side a is increasing at a rate of 1.5 meters/second and c is fixed at 8 meters.
    i) Find the rate of change of the measure of angle A when b=4 meters.
    ii) Find the rate of change of the measure of angle B when b=4 meters.
    Last edited by Velvet Love; October 16th 2009 at 10:02 AM. Reason: Forgot picture
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  2. #2
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    We first write everything in mathematical notation:

    \frac{da}{dt}=1.5\;\;\;\;\;\;\;\;\;\;c=8 \;\;\;\;\;\;\;\;\;\;\sin A=\cos B=\frac{a}{c}\;\;\;\;\;\;\;\;\;\;\cos A=\sin B=\frac{b}{c}

    Now we may use the Chain Rule, remembering that a, b, and A are all functions of t.
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  3. #3
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    Would the answer be sin(sqrt(48)/8) * 1.5? For the first one
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  4. #4
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    Quote Originally Posted by Velvet Love View Post
    Would the answer be sin(sqrt(48)/8) * 1.5? For the first one
    For the answer to the first, we may differentiate both sides of \sin A=\frac{a}{c}:

    \begin{aligned}<br />
\frac{d}{dt}(\sin A)&=\frac{d}{dt}\left(\frac{a}{c}\right)\\<br />
\cos A\,\frac{dA}{dt}&=\frac{1}{c}\frac{da}{dt}=\frac{1  .5}{8}.<br />
\end{aligned}

    Now we may solve for \frac{dA}{dt} knowing the value of \cos A.
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  5. #5
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    so cosA is cos(sqrt(48)/8)
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  6. #6
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    Not quite. From the definition of cosine, we know that

    \cos A=\frac{b}{c}.
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  7. #7
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    Oh ok. I think i got it now.
    It should be (1.5/8)/(cos(4/8)) right?
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  8. #8
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    Quote Originally Posted by Velvet Love View Post
    Oh ok. I think i got it now.
    It should be (1.5/8)/(cos(4/8)) right?
    Not quite. We know that c=8 from the statement of the problem. When b=4, what is \frac{b}{c}?

    Here is the next step:

    <br />
\begin{aligned}<br />
\cos A\,\frac{dA}{dt}&=\frac{1.5}{8}\\<br />
\frac{b}{c} \frac{dA}{dt}&=\frac{1.5}{8}\\<br />
\frac{c}{b}\cdot\frac{b}{c}\frac{dA}{dt}&=\frac{c}  {b}\cdot\frac{1.5}{8}\\<br />
\frac{dA}{dt}&=\frac{c}{b}\cdot\frac{1.5}{8}.<br />
\end{aligned}<br />
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  9. #9
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    4/8 or 1/2
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  10. #10
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    If the first answer is (3/8), then the second one is -(3/8) right?
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  11. #11
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    Quote Originally Posted by Velvet Love View Post
    4/8 or 1/2
    Correct. The value of \frac{dA}{dt} is therefore

    \frac{dA}{dt}=\frac{2}{1}\cdot\frac{1.5}{8}.
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