Question

A physical quantity X is related to four measurable quantities a, b, c and d as follows: X = (a^2)(b^3)(c^5/2)(d^–2). The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity X ?
​

Answer 1

Answer:

X=a  

2

b  

3

c  

5/2

d  

−2

 

Percentage error in X is

x

ΔX

​

×100=[2(  

a

Δa

​

)+3(  

b

Δb

​

)+  

2

5

​

(  

c

Δc

​

)+2(  

d

Δd

​

)]×100

=2×1+3×2+  

2

5

​

×2+2×4

=21%

Explanation:

Answer 2

Given:

• A physical quantity is related to four measurable quantities a, b, c and d.

• The percentage error in quantities a, b, c and d are 1%, 2%, 3%, and 4% respectively.

To Find:

• The percentage error in quantity X.

Solution:

The quantity X is as follows:

• $\sf{X = \dfrac{a^2 b^3 c^{5/2}}{2 d^2}}$

The Percentage error in the given measurable quantities are as follows:

• $\sf{\dfrac{\Delta a}{a} \times 100 = 1\%}$

• $\sf{\dfrac{\Delta b}{b} \times 100 = 2\%}$

• $\sf{\dfrac{\Delta c}{c} \times 100 = 3\%}$

• $\sf{\dfrac{\Delta d}{d} \times 100 = 4\%}$

Now the error in quantity X will be as following:

$\sf{\dfrac{\Delta X}{X} = \pm \bigg\{2\times \dfrac{\Delta a}{a} + 3\times \dfrac{\Delta b}{b} + \dfrac{5}{2}\times \dfrac{\Delta c}{c} + 2\times \dfrac{\Delta d}{d}\bigg\}}$

Now Percentage error in quantity X is as follows:

$= \sf{\dfrac{\Delta X}{X} \times 100 = \pm \bigg\{\bigg(2\times \dfrac{\Delta a}{a} \times 100\bigg) + \bigg(3 \times \dfrac{\Delta b}{b} \times 100\bigg) + \bigg(\dfrac{5}{2} \times \dfrac{\Delta c}{c} \times 100 \bigg) + \bigg(2 \times \dfrac{\Delta d}{d} \times 100\bigg)\bigg\}}$

Putting the values of (∆a/a × 100), (∆b/b × 100), (∆c/c × 100) and (∆d/d × 100):

$= \sf{\dfrac{\Delta X}{X} \times 100 = \pm \bigg\{(2 \times 1) + (3 \times 2) + \bigg(\dfrac{5}{2} \times 3 \bigg) + (2 \times 4)\bigg\}}$

$= \sf{\dfrac{\Delta X}{X} \times 100 = \pm\{ 2 + 6 + (2.5 \times 3) + 8\}}$

$= \sf{\%\:error\:in\:X = \pm\{ 16 + 7.5\}}$

$= \sf{\%\:error\:in\:X = \pm 23.5\%}$

∴ The percentage error in the given quantity X is ±23.5%

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