Question

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

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

Step3: Evaluate \(\sqrt{2}\)

\(\sqrt{2}\approx1.414\), a positive number greater than \(1\).

Step4: Compare the values

Negative numbers are less than positive numbers. So \(5 - \pi^2\) (negative) is less than \(\sqrt{\frac{1}{2}}\) (positive, \(\approx0.707\)) and \(\sqrt{\frac{1}{2}}\) is less than \(\sqrt{2}\) (\(\approx1.414\)). So the order from least to greatest is \(5-\pi^2,\sqrt{\frac{1}{2}},\sqrt{2}\).

Answer:

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