To evaluate \( 3\sqrt{0.0072 \times 81.3^2} \) using logarithm tables, we can follow these steps:
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Break down the expression: We can rewrite the expression under the square root: \[ 3\sqrt{0.0072 \times 81.3^2} = 3 \times \sqrt{0.0072} \times \sqrt{81.3^2} \] Since \(\sqrt{81.3^2} = 81.3\), we can further simplify: \[ = 3 \times 81.3 \times \sqrt{0.0072} \]
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Calculate \( \sqrt{0.0072} \): To find \(\sqrt{0.0072}\), we can express it in terms of its logarithm. We know that: \[ 0.0072 = 7.2 \times 10^{-3} \] Therefore: \[ \log_{10}(0.0072) = \log_{10}(7.2) + \log_{10}(10^{-3}) = \log_{10}(7.2) - 3 \]
Consulting a logarithm table, we find: \[ \log_{10}(7.2) \approx 0.857 \] Thus: \[ \log_{10}(0.0072) = 0.857 - 3 = -2.143 \]
To find \(\sqrt{0.0072}\): \[ \log_{10}(\sqrt{0.0072}) = \frac{1}{2} \log_{10}(0.0072) = \frac{1}{2} \times (-2.143) \approx -1.0715 \] This implies: \[ \sqrt{0.0072} \approx 10^{-1.0715} \approx 0.084 \]
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Calculate \( 3 \times 81.3 \times \sqrt{0.0072} \): Now we know: \[ 3 \times 81.3 \approx 243.9 \] Multiplying: \[ 243.9 \times \sqrt{0.0072} \approx 243.9 \times 0.084 \approx 20.49 \]
So the final evaluation of \( 3\sqrt{0.0072 \times 81.3^2} \) is approximately:
\[ \boxed{20.49} \]