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: The relationship between P and a, b, c, d
The relationship can be written as:
P = a³ b² c¹ d^-1/2

Step 2: The formula for maximum percentage error
The maximum percentage error in P is given by the sum of the absolute values of the percentage errors in the individual quantities multiplied by their respective powers:
(ΔP / P) × 100% = |3 × (Δa / a × 100%)| + |2 × (Δb / b × 100%)| + |1 × (Δc / c × 100%)| + |(-1/2) × (Δd / d × 100%)|

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

Step 4: Simplifying the expression
Now, simplifying each term:
(ΔP / P) × 100% = |3%| + |6%| + |2%| + |-2%|

Step 5: Adding the values
Add the values together:
(ΔP / P) × 100% = 3% + 6% + 2% + 2%

Step 6: Final result
The final result is:
(ΔP / P) × 100% = 13%