To determine which number results in an irrational number when multiplied by 0.4, let's analyze each choice step by step and check if the product is irrational.
### Choice A: [tex]\( 3 \pi \)[/tex]
- Calculate [tex]\( 3 \pi \times 0.4 \)[/tex]:
[tex]\[
3 \pi \times 0.4 \approx 3.7699111843077517
\][/tex]
- Since [tex]\(\pi\)[/tex] is an irrational number, [tex]\(3 \pi\)[/tex] remains irrational. When an irrational number is multiplied by a rational number, the result is also irrational. Therefore:
[tex]\[
3 \pi \times 0.4 = 3.7699111843077517 \, \text{(Irrational)}
\][/tex]
### Choice B: [tex]\( \sqrt{9} \)[/tex]
- Simplify [tex]\( \sqrt{9} \)[/tex]:
[tex]\[
\sqrt{9} = 3
\][/tex]
- Calculate [tex]\( 3 \times 0.4 \)[/tex]:
[tex]\[
3 \times 0.4 = 1.2000000000000002
\][/tex]
- 1.2 is a rational number because it can be expressed as a fraction [tex]\(\frac{6}{5}\)[/tex]. Therefore:
[tex]\[
3 \times 0.4 = 1.2000000000000002 \, \text{(Rational)}
\][/tex]
### Choice C: [tex]\( \frac{2}{7} \)[/tex]
- Calculate [tex]\( \frac{2}{7} \times 0.4 \)[/tex]:
[tex]\[
\frac{2}{7} \times 0.4 \approx 0.11428571428571428
\][/tex]
- 0.11428571428571428 can be expressed as a repeating decimal, which indicates it is a rational number. Therefore:
[tex]\[
\frac{2}{7} \times 0.4 = 0.11428571428571428 \, \text{(Rational)}
\][/tex]
### Choice D: [tex]\( 0.444 \ldots \)[/tex]
- Calculate [tex]\( 0.444 \ldots \times 0.4 \)[/tex]:
[tex]\[
0.444 \ldots \times 0.4 \approx 0.1777776
\][/tex]
- 0.1777776 is a repeating decimal and can be expressed as a rational number [tex]\(\frac{8}{45}\)[/tex]. Therefore:
[tex]\[
0.444 \ldots \times 0.4 = 0.1777776 \, \text{(Rational)}
\][/tex]
From these computations, only [tex]\(3 \pi \times 0.4\)[/tex] results in an irrational number. Therefore, the answer is:
[tex]\[
\boxed{A. \, 3\pi}
\][/tex]
