Answer :
Answer:
To find the value of \( a \), we need to analyze the given equation for the physical quantity \( P \) and calculate the percentage error.
### Given:
\[ P = \frac{\sqrt{ab} \cdot d^a}{\sqrt{c}} \]
The percentage errors in \( a \), \( b \), \( c \), and \( d \) are all 0.5%.
The percentage error in \( P \) is 2%.
### Step-by-Step Solution:
First, let's rewrite \( P \) in a more convenient form for error analysis.
\[ P = \frac{(ab)^{1/2} \cdot d^a}{c^{1/2}} \]
Taking the natural logarithm on both sides to make differentiation easier:
\[ \ln P = \ln \left( \frac{(ab)^{1/2} \cdot d^a}{c^{1/2}} \right) \]
Using logarithm properties:
\[ \ln P = \ln \left( (ab)^{1/2} \right) + \ln \left( d^a \right) - \ln \left( c^{1/2} \right) \]
\[ \ln P = \frac{1}{2} \ln (ab) + a \ln d - \frac{1}{2} \ln c \]
Further expanding the logarithms:
\[ \ln P = \frac{1}{2} (\ln a + \ln b) + a \ln d - \frac{1}{2} \ln c \]
Differentiating both sides to find the relative error:
\[ \frac{dP}{P} = \frac{1}{2} \left( \frac{da}{a} + \frac{db}{b} \right) + a \frac{dd}{d} - \frac{1}{2} \frac{dc}{c} \]
The relative errors of \( a \), \( b \), \( c \), and \( d \) are given as 0.5% each. Converting to decimal form for calculation:
\[ \frac{da}{a} = \frac{db}{b} = \frac{dc}{c} = \frac{dd}{d} = 0.005 \]
Substituting these into the error equation:
\[ \frac{dP}{P} = \frac{1}{2} (0.005 + 0.005) + a \cdot 0.005 - \frac{1}{2} \cdot 0.005 \]
Simplifying:
\[ \frac{dP}{P} = \frac{1}{2} \cdot 0.01 + a \cdot 0.005 - 0.0025 \]
\[ \frac{dP}{P} = 0.005 + 0.005a - 0.0025 \]
\[ \frac{dP}{P} = 0.0025 + 0.005a \]
Given the percentage error in \( P \) is 2% or 0.02 in decimal:
\[ 0.02 = 0.0025 + 0.005a \]
Solving for \( a \):
\[ 0.02 - 0.0025 = 0.005a \]
\[ 0.0175 = 0.005a \]
\[ a = \frac{0.0175}{0.005} \]
\[ a = 3.5 \]
Therefore, the value of \( a \) is:
\[ \boxed{3.5} \]
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Answer:
2)
2/5
Explanation:
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