Which irrational number lies between the rational numbers 109 and 110? a) 5π b) √222 c) 10√3 d) (√2)³
The Correct Answer and Explanation is:
To find an irrational number that lies between the rational numbers 109 and 110, we will evaluate each of the given options: (5\pi), (\sqrt{222}), (10\sqrt{3}), and ((\sqrt{2})^3).
- Calculating (5\pi):
[
5\pi \approx 5 \times 3.14 = 15.7
]
This value is far greater than 110 and does not lie between 109 and 110. - Calculating (\sqrt{222}):
To determine if (\sqrt{222}) is between 109 and 110, we first compute:
[
109^2 = 11881 \quad \text{and} \quad 110^2 = 12100
]
Since (11881 < 222 < 12100), we conclude that:
[
109 < \sqrt{222} < 110
]
Therefore, (\sqrt{222}) lies between 109 and 110. - Calculating (10\sqrt{3}):
[
10\sqrt{3} \approx 10 \times 1.732 = 17.32
]
This value is also significantly greater than 110, ruling it out. - Calculating ((\sqrt{2})^3):
[
(\sqrt{2})^3 = 2\sqrt{2} \approx 2 \times 1.414 = 2.828
]
This value is also much less than 109, so it does not qualify.
After evaluating each option, we find that (\sqrt{222}) is the only number that fits the criteria of being irrational and lying between 109 and 110.
Conclusion
The correct answer is b) (\sqrt{222}), as it is the only irrational number among the options provided that lies in the specified range. Rational numbers are numbers that can be expressed as a fraction of two integers, while irrational numbers cannot be expressed this way. The approximation and inequalities demonstrate that (\sqrt{222}) is an irrational number between 109 and 110, fulfilling the problem’s requirements.