Maths Help

Question

A right triamgle is 20cm long on one side and the opposite angle is 30 degrees with a possible error of +or- 1 degree

a) estimate the error in computing the length of the hypotenuse

Please give full solution

Answer 1

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Calculated length L = 20cm/sin30 = 40cm

Possible error of angle is +/- 1 degree

Max of L will be 20cm/sin29 = 41.25cm = 40cm + 1.25cm

Min of L will be 20cm/sin31 = 38.83cm = 40cm - 1.17cm

The error is +1.25 -1.17

When the error in angle is small compared with the absolute value, a usual way to estimate error is by means of differentiation.

Since L = 20 / sinX where X is in radian

dL/dX = -20cosX/sin^2X

dL = -20 cosX/sin^2X(dX)

dL = -20 cos(30)/sin^2(30)(+/-2丌/360)

dL = +/- 20 * 0.866 * 4 * 2丌/360

approximately equals = +/-1.2

When the error in angle is smaller this approximation will be more accurate compared with actual.