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If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4?
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If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,...
- **Calculating Percentage Error in z**
Calculating the percentage error in z requires understanding how the errors in A, B, C, and D propagate through the expression z = A^2 * B^(1/2) / C^(1/3) * D^(1/4).
- **Error Propagation Formula**
The formula for calculating the percentage error in z given the errors in A, B, C, and D is:
\[ \frac{\Delta z}{z} = \sqrt{(\frac{\Delta A}{A})^2 + (\frac{\Delta B}{B})^2 + (\frac{\Delta C}{C})^2 + (\frac{\Delta D}{D})^2 } \]
- **Substitute the Given Errors**
Substitute the given percentage errors into the formula:
\[ \frac{\Delta z}{z} = \sqrt{(0.01)^2 + (0.02)^2 + (0.03)^2 + (0.04)^2 } \]
- **Calculate the Percentage Error**
Calculating the percentage error:
\[ \frac{\Delta z}{z} = \sqrt{0.0001 + 0.0004 + 0.0009 + 0.0016 } \]
\[ \frac{\Delta z}{z} = \sqrt{0.003 } \]
\[ \frac{\Delta z}{z} = 0.055 \]
- **Conclusion**
The percentage error in the measurement of z is approximately 5.5%. This means that the measurement of z is sensitive to errors in A, B, C, and D, and the overall uncertainty in z is 5.5%.
Calculating the percentage error in z requires understanding how the errors in A, B, C, and D propagate through the expression z = A^2 * B^(1/2) / C^(1/3) * D^(1/4).
- **Error Propagation Formula**
The formula for calculating the percentage error in z given the errors in A, B, C, and D is:
\[ \frac{\Delta z}{z} = \sqrt{(\frac{\Delta A}{A})^2 + (\frac{\Delta B}{B})^2 + (\frac{\Delta C}{C})^2 + (\frac{\Delta D}{D})^2 } \]
- **Substitute the Given Errors**
Substitute the given percentage errors into the formula:
\[ \frac{\Delta z}{z} = \sqrt{(0.01)^2 + (0.02)^2 + (0.03)^2 + (0.04)^2 } \]
- **Calculate the Percentage Error**
Calculating the percentage error:
\[ \frac{\Delta z}{z} = \sqrt{0.0001 + 0.0004 + 0.0009 + 0.0016 } \]
\[ \frac{\Delta z}{z} = \sqrt{0.003 } \]
\[ \frac{\Delta z}{z} = 0.055 \]
- **Conclusion**
The percentage error in the measurement of z is approximately 5.5%. This means that the measurement of z is sensitive to errors in A, B, C, and D, and the overall uncertainty in z is 5.5%.
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If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4?
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If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4?.
If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If percentage error in measurement of quantities A,B,C,D are 1%,2%,3%,4% respectively the percentage error in measurement of z=A^2B^1/2/C^1/3D^1/4?.
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