Question

which set is ordered from least to greatest?
a. $\sqrt{2}$, $\sqrt{\frac{1}{2}}$, $5 - \pi^2$
b. $\sqrt{\frac{1}{2}}$, $\sqrt{2}$, $5 - \pi^2$
c. $5 - \pi^2$, $\sqrt{2}$, $\sqrt{\frac{1}{2}}$
d. $5 - \pi^2$, $\sqrt{\frac{1}{2}}$, $\sqrt{2}$

Explanation:

Step1: Evaluate \(5 - \pi^2\)

We know that \(\pi\approx3.14\), so \(\pi^2\approx9.86\). Then \(5 - \pi^2\approx5 - 9.86=- 4.86\), which is a negative number.

Step2: Evaluate \(\sqrt{\frac{1}{2}}\)

\(\sqrt{\frac{1}{2}}=\frac{\sqrt{2}}{2}\approx\frac{1.414}{2} = 0.707\), which is a positive number less than \(\sqrt{2}\) (since \(\sqrt{2}\approx1.414\)).

Step3: Compare the three numbers

We have \(5 - \pi^2\approx - 4.86\), \(\sqrt{\frac{1}{2}}\approx0.707\), and \(\sqrt{2}\approx1.414\). So the order from least to greatest is \(5 - \pi^2\), \(\sqrt{\frac{1}{2}}\), \(\sqrt{2}\).

Answer:

D. \(5 - \pi^2\), \(\sqrt{\frac{1}{2}}\), \(\sqrt{2}\)