Question

A physical quantity P is related to four observations a, b, c, and d as follows: P = a³ b² (c / √d) The percentage errors of measurement in a, b, c, and d are 1%, 3%, 2%, and 4% respectively. The percentage error in the quantity P is:

• A. \( 2\% \)
• B. \( 13\% \)
• C. \( 15\% \)
• D. \( 10\% \)

Hint:

Remember that for quantities related by multiplication and division, the percentage errors add up (after multiplying by their powers). The power of a term in the denominator becomes negative, but we take the absolute value when calculating the maximum percentage error.

Answer 1

The Correct Option is B

Step 1: Defining the relationship between P and its variables
The relationship is defined as:
P = a³ b² c¹ d^-1/2

Step 2: Formulating the maximum percentage error
The maximum percentage error in P is calculated by summing the absolute values of the percentage errors of individual quantities, each multiplied by its respective exponent:
(ΔP / P) × 100% = |3 × (Δa / a × 100%)| + |2 × (Δb / b × 100%)| + |1 × (Δc / c × 100%)| + |(-1/2) × (Δd / d × 100%)|

Step 3: Applying the given percentage errors
Substitute the provided percentage errors for each variable:
(ΔP / P) × 100% = |3 × 1%| + |2 × 3%| + |1 × 2%| + |(-1/2) × 4%|

Step 4: Simplifying the error components
Simplify each term in the expression:
(ΔP / P) × 100% = |3%| + |6%| + |2%| + |-2%|

Step 5: Aggregating the error contributions
Sum the simplified error values:
(ΔP / P) × 100% = 3% + 6% + 2% + 2%

Step 6: Presenting the final maximum percentage error
The total maximum percentage error in P is:
(ΔP / P) × 100% = 13%