# Thread: Estimate relative error using differentials

1. ## 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.

2. Originally Posted by tillymc
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 $\displaystyle \theta$ is equal to 30 degrees, then $\displaystyle A = 2\sqrt{3} \, cm^2$

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

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

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

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

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

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

$\displaystyle dA = \frac{\pi}{180}$

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