Question

The quantity x depends on a, b as x=2a^2/b If error in measurement of a, b is 1% and 2% respectively, then error in measurement of x, is

Answer 1

Given :- The quantity x depends on a, b as x=2a^2/b If error in measurement of a, b is 1% and 2% respectively, then error in measurement of x, is ?

Solution :-

we have,

→ x = 2a²/b

so,

→ ∆x/x = 2(∆a/a) + ∆b/b

given that, error in measurement of a, b is 1% and 2% respectively . so,

• ∆a/a = 1%
• ∆b/b = 2%

so,

→ ∆x/x = 2 * 1 + 2

→ ∆x/x = 2 + 2

→ ∆x/x = 4 %

therefore, the % error in measurement of x is 4% .

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Answer 2

Given:

The quantity x depends on a, b as x=2a²/b. The error in measurement of a, b is 1% and 2% respectively.

To find:

Error in measurement of x ?

Calculation:

$x = \dfrac{2 {a}^{2} }{b}$

• If you see the method of error calculation, it mainly involves taking logarithms and then differentiation (i.e. calculus).

• So, whenever there is a physical quantity raised to a certain exponential, after taking logarithm, the exponential value gets multiplied.

• For example : If Z = A^(n), so we will say ∆Z/Z = n × (∆A/A).

• And for max errors, individual errors always have to be added.

Now, for small changes in values of a and b (< 6%), we can say:

$\implies\dfrac{\Delta x}{x} = \bigg( 2 \times \dfrac{\Delta a}{a} \bigg) + \dfrac{\Delta b}{b}$

$\implies\dfrac{\Delta x}{x} = \bigg( 2 \times 1\%\bigg) + 2\%$

$\implies\dfrac{\Delta x}{x} = 2\% + 2\%$

$\implies\dfrac{\Delta x}{x} = 4\%$

So % error in measurement of x is 4%.