Source: Extra Practice

A physical quantity XX is given by X=A2B3CDX = \frac{A^2 B^3}{C \sqrt{D}}. If the percentage errors in the measurements of A,B,C,A, B, C, and DD are 1%,2%,3%1\%, 2\%, 3\%, and 4%4\% respectively, what is the total percentage error in XX?

Options

Option A is correct

13%13\%

Option B

14%14\%

Option C

15%15\%

Option D

16%16\%

Explanation

For a quantity X=ApBqCrDsX = \frac{A^p B^q}{C^r D^s}, the maximum percentage error in XX is given by: \%E_X = p(\text{%E}_A) + q(\text{%E}_B) + r(\text{%E}_C) + s(\text{%E}_D). In the given equation, X=A2B3C−1D−1/2X = A^2 B^3 C^{-1} D^{-1/2}. Note that the powers are always taken as positive when summing errors. \%E_X = 2(\text{%E}_A) + 3(\text{%E}_B) + 1(\text{%E}_C) + \frac{1}{2}(\text{%E}_D). Given percentage errors: \text{%E}_A = 1\% \text{%E}_B = 2\% \text{%E}_C = 3\% \text{%E}_D = 4\% Substitute these values into the formula: %EX=2(1%)+3(2%)+1(3%)+12(4%)\%E_X = 2(1\%) + 3(2\%) + 1(3\%) + \frac{1}{2}(4\%) %EX=2%+6%+3%+2%\%E_X = 2\% + 6\% + 3\% + 2\% %EX=13%\%E_X = 13\%.