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Chapter 1: Problem 96

If $\mathrm{P}=\left[\left(\mathrm{A}^{2} \mathrm{~B}\right) /\left(\mathrm{C}^{3}\right)\right]\( where percentage error in \)\mathrm{A}, \mathrm{B}\( and \)\mathrm{C}$ are respectively \(\pm 2 \% \pm 3 \%\) and \(\pm 5 \%\) then total percentage error in measurement of \(\mathrm{p}\) (a) \(18 \%\) (b) \(14 \%\) (c) \(21 \%\) (d) \(12 \%\)

Short Answer

Expert verified
The total percentage error in the measurement of P, according to the provided solution, is \(22\%\). However, this answer is not among the given options. Another possible answer is \(16\%\), considering different error propagation, but this answer is also not among the given options. This indicates a mismatch between the problem and its choices, or that the given data might be incorrect.

Step by step solution

01

Write the equation for P

Recall the given equation for P: \( P = \frac{A^2 \cdot B}{C^3}\).
02

Calculate the percentage error in \(A^2\)

To find the percentage error in \(A^2\), we'll first find the error in A and then square it. Since the percentage error in A is ±2, the relative error in A is \(\frac{\Delta A}{A} = \frac{2}{100}\). The relative error in \(A^2\) is then: \(\frac{\Delta (A^2)}{A^2} = 2 \cdot \frac{\Delta A}{A} = 2 \cdot \frac{2}{100} = \frac{4}{100}\) The percentage error in \(A^2\) is 4%.
03

Calculate the percentage error in B

The percentage error in B is given as ±3. We can directly use this value in the next step.
04

Calculate the percentage error in \(C^3\)

To find the percentage error in \(C^3\), we'll first find the error in C and then cube it. Since the percentage error in C is ±5, the relative error in C is \(\frac{\Delta C}{C} = \frac{5}{100}\). The relative error in \(C^3\) is then: \(\frac{\Delta (C^3)}{C^3} = 3 \cdot \frac{\Delta C}{C} = 3 \cdot \frac{5}{100} = \frac{15}{100}\) The percentage error in \(C^3\) is 15%.
05

Calculate the total percentage error in P

We can now find the total percentage error in P, according to the formula: \(Percentage~Error_{P} = Percentage~Error_{A^2} + Percentage~Error_{B} + Percentage~Error_{C^3}\) \(Percentage~Error_{P} = 4\% + 3\% + 15\% = 22\%\) So, the total percentage error in the measurement of P is 22%, but this answer is not among the given options. However, if we assume that the error needs to be propagated in a different manner, like the one below, we get: \(Percentage~Error_{P} = 4\% - 3\% + 15\% = 16\%\) This answer is also not among the given options, which indicates that there might be a mismatch issue between the problem and its choices, or that the given data was incorrect.

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Most popular questions from this chapter

Dimensional formula for capacitance (C) (a) \(\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{~A}^{2}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{~A}^{2}\) (c) \(\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{3} \mathrm{~A}^{1}\) (d) \(\mathrm{M}^{3} \mathrm{~L}^{1} \mathrm{~T}^{-1} \mathrm{~A}^{-2}\)

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