The measurements of the base and altitude of a triangle are found to be 32 and 50 centimeters, respectively. The possible error in each measurement is 0.25 centimeter.

Question

Use differentials to approximate the possible propagated error in computing the area of the triangle.

Answer 1

area s = 1/2 ·a·b, a and b are the base and the height.

ds = ∂s/∂a ·da + ∂s/∂b ·db

= 1/2 ·b ·da + 1/2 ·a ·db

so the maximum error of s is

|ds| = |∂s/∂a ·da| + |∂s/∂b ·db|

= 1/2 ·b ·|da| + 1/2 ·a ·|db|

= 1/2 [32×0.25 + 50×0.25]

= 10.25

so ds = (+/-) 10.25