To determine which number produces a rational number when multiplied by 0.5, we need to check each option individually and determine whether the result is rational. A rational number can be expressed as a fraction of two integers.
Let's analyze each option:
A. [tex]\(-1.73205089 \ldots\)[/tex]
If we multiply [tex]\(-1.73205089 \ldots\)[/tex] by 0.5:
[tex]\[ -1.73205089 \ldots \times 0.5 = -0.866025445 \ldots \][/tex]
This result is a non-repeating, non-terminating decimal, thus it is an irrational number.
B. [tex]\(\sqrt{3}\)[/tex]
If we multiply [tex]\(\sqrt{3}\)[/tex] by 0.5:
[tex]\[ \sqrt{3} \times 0.5 = \frac{\sqrt{3}}{2} \][/tex]
Since [tex]\(\sqrt{3}\)[/tex] is irrational, [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is also irrational.
C. [tex]\(0.54732814 \ldots\)[/tex]
If we multiply [tex]\(0.54732814 \ldots\)[/tex] by 0.5:
[tex]\[ 0.54732814 \ldots \times 0.5 = 0.27366407 \ldots \][/tex]
This result is a non-repeating, non-terminating decimal, thus it is an irrational number.
D. [tex]\(\frac{1}{3}\)[/tex]
If we multiply [tex]\(\frac{1}{3}\)[/tex] by 0.5:
[tex]\[ \frac{1}{3} \times 0.5 = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \][/tex]
The fraction [tex]\(\frac{1}{6}\)[/tex] is a ratio of two integers (1 and 6), so it is a rational number.
After evaluating each option, we find that the number which produces a rational number when multiplied by 0.5 is:
[tex]\[ \boxed{D. \frac{1}{3}} \][/tex]
