Use differentials to estimate the relative errors of the area A if H=4 exactly and theta is measured to be 30 deg with measurement of 15 minutes o arcThe area of a right triangle with hypotenuse H...

Use differentials to estimate the relative errors of the area A if H=4 exactly and theta is measured to be 30 deg with measurement of 15 minutes o arc

The area of a right triangle with hypotenuse H is given by the formula

A=(1/4)H^2sin(2theta)  where theta is one of the acute angles.

Asked on by bdremm

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lfryerda | High School Teacher | (Level 2) Educator

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We can approximate the formula `A=1/4h^2sin(2 theta)` using differentials to get

`dA=1/2h dh sin(2 theta)+1/2 h^2 cos(2 theta) d theta`

Since h=4 is exact, this means that `dh=0` , so the first term on the right side vanishes.  On the other hand, `theta=30^circ=pi/6` with an error of 15 minutes of arc, which is the same as `d theta=15/60 ^circ=1/4 ^circ=pi/720`

The angle must be put in radians or the differentials are incorrect.

Now we can substitute into the formula for the differential area to get

`dA=1/2h^2cos(2 theta) d theta`

`=1/2(4^2) cos(pi/3)(pi/720)`

`=pi/90(1/2)`

`=pi/180`

`approx 0.0175` units squared.

The area has an error of approximately 0.0175 units squared.

To find the relative error, we need to calculate the area suing the formula:

`A=1/4h^2sin(2 theta)=1/4 16 sin(pi/3) approx 3.46` units squared

so the relative error is

`epsilon=0.0175/3.46 times 100% approx 0.49%`

The relative error is approximately 0.49%

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