Question

Evaluate using logarithm table: 3√0.0072×81.3^2

Answer 1

To evaluate \( 3\sqrt{0.0072 \times 81.3^2} \) using logarithm tables, we can follow these steps:

1. **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:
\[
= 3 \times 81.3 \times \sqrt{0.0072}
\]

2. **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:
\[
\sqrt{0.0072} \approx 10^{-1.0715} \approx 0.084
\]

3. **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:

\[
\boxed{20.49}
\]