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.
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`
`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%