To solve the equation \( \frac{x^2}{0.3} = 64 \), we first isolate \( x^2 \):
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Multiply both sides by 0.3: \[ x^2 = 64 \times 0.3 \]
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Calculate \( 64 \times 0.3 \): \[ 64 \times 0.3 = 19.2 \] So we have: \[ x^2 = 19.2 \]
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Now take the square root of both sides: \[ x = \sqrt{19.2} \]
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To simplify \( \sqrt{19.2} \), express 19.2 as a fraction: \[ 19.2 = \frac{192}{10} = \frac{96}{5} \]
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Now let's find: \[ x = \sqrt{\frac{96}{5}} = \frac{\sqrt{96}}{\sqrt{5}} = \frac{4\sqrt{6}}{\sqrt{5}} = \frac{4\sqrt{30}}{5} \]
Now, we can calculate the decimal value:
- First, find the estimated value of \( \sqrt{30} \) which is approximately \( 5.477 \).
- Thus: \[ x \approx \frac{4 \times 5.477}{5} \approx \frac{21.908}{5} \approx 4.3816 \]
Since none of the exact values are integer values, let's compare \( \sqrt{19.2} \) with the options given.
We can check each of the provided answers to find the one that is closest to \( 4.3816 \):
- 32
- 32
- 16
- 165
- 125
- 128
None of the direct comparisons match, but it's likely there is an error in interpreting or calculating the options.
To estimate:
- 16 is the closest integer and is a value that can be considered within the rational interpretation of the value of \( x \).
If we pick amongst the integers, the answer to pick would be 16 as it fits as the nearest result when considering the roots.
Thus the best response to choose from the provided options is 16.