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Math Help - Estimate relative error using differentials

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    Estimate relative error using differentials

    Area of right triangle with hypotenuse H is

    A=(1/4)H^2sin(2theta)

    where theta is one of the acute angles.

    Use differentials to estimate the relative errors of the area A if H=4cm and theta is measured to be 30 degrees with an error of measurement of 15 minutes of arc.

    note: a minute of arc, 1' is approximately equal to (1/60) of a degree.

    I'm not quite sure what a measurement of arc is and whether i'm supposed to find the derivative or what? Im so lost, thanks in advance.
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    Quote Originally Posted by tillymc View Post
    Area of right triangle with hypotenuse H is

    A=(1/4)H^2sin(2theta)

    where theta is one of the acute angles.

    Use differentials to estimate the relative errors of the area A if H=4cm and theta is measured to be 30 degrees with an error of measurement of 15 minutes of arc.

    note: a minute of arc, 1' is approximately equal to (1/60) of a degree.

    I'm not quite sure what a measurement of arc is and whether i'm supposed to find the derivative or what? Im so lost, thanks in advance.

    if \theta is equal to 30 degrees, then A = 2\sqrt{3} \, cm^2

    note that 15' = \left(\frac{15}{60}\right)^\circ = \frac{\pi}{720} \, rad

    radians need to be used since the derivative of sine = cosine only works in radians.

    A = \frac{H^2}{4} \sin(2\theta)

    \frac{dA}{d\theta} = \frac{H^2}{2} \cos(2\theta)<br />

    dA = \frac{H^2}{2} \cos(2\theta) \, d\theta

    dA = \frac{4^2}{2} \cos\left(\frac{\pi}{3}\right) \cdot \frac{\pi}{720}

    dA = \frac{\pi}{180}

    \frac{dA}{A} \approx 0.005 ... area error is about 0.5%
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