Question

The length and breadth of a rectangle are measured as ( a +- delta) and ( b+-deltab) respectively find (1) relative error (2) absolute error in measurement of area​

Answer 1

(i) Length of the rectangle, L = (a ± ∆a)

breadth of the rectangle , B = (b ± ∆b)

we know, area of rectangle , A = L × B

taking log both sides,

logA = log(L × B) = logL + logB

now differentiating both sides,

dA/A = dL/L + dB/B

if dA comparable to A then, dA → ∆A

similarly, dB → ∆B and dL → ∆L

then, ∆A/A = ∆L/L + ∆B/B

here, ∆L = ∆a, L =a, ∆B = ∆b and B = b

∆A/A = ∆a/a + ∆b/b

we know, relative error is the ratio of absolute error to original value.

so, relative error of area = ∆a/a + ∆b/b

(ii) ∆A is absolute error of area.

here A = a × b [ as L = a , and B = b]

so, ∆A/ab = ∆a/a + ∆b/b

or, ∆A = ab(∆a/a + ∆b/b)

or, ∆A = b.∆a + a.∆b

hence, absolute error of area = b.∆a + a.∆b