Question

The air bubble formed by explosion inside water perform oscillations with time

period X is connected from X=ab2

C. Calculate the percentage error in X, when % error

in a, b, c are 4, 2, and 3 respectively.

Answer 1

given that the time period is

$X = ab^2c$

now the percentage error in a,b and c is given as

%a = 4%

%b = 2%

%c = 3%

now for the percentage error in X is given by

%X = %a + 2%b + %c

it is derived by following method

first we will take log on both sides of equation

$ln X = ln a + 2ln b + ln c$

now differentiate whole equations

$\frac{dx}{x} = \frac{da}{a} + 2\frac{db}{b} + \frac{dc}{c}$

so now we will plug in all data in above equation

$X = 4 + 2*2 + 3$

%X = 11%

so percentage error in X will be 11%