Question

a number line is shown.

which values are ordered least to greatest when plotted on the number line?

a. $0.5\sqrt{170}$, $\sqrt{120}$, $2\sqrt{50}$

b. $\sqrt{120}$, $0.5\sqrt{170}$, $2\sqrt{50}$

c. $2\sqrt{50}$, $\sqrt{120}$, $0.5\sqrt{170}$

d. $\sqrt{120}$, $2\sqrt{50}$, $0.5\sqrt{170}$

Explanation:

Step1: Simplify each expression

• For \(0.5\sqrt{170}\):

First, calculate \(0.5\sqrt{170}=\frac{\sqrt{170}}{2}\). Then, \(\sqrt{170}\approx13.04\), so \(\frac{\sqrt{170}}{2}\approx\frac{13.04}{2} = 6.52\).

• For \(\sqrt{120}\):

\(\sqrt{120}\approx10.95\) (since \(10^2 = 100\), \(11^2=121\), so it's between 10 and 11, closer to 10.95).

• For \(2\sqrt{50}\):

Simplify \(2\sqrt{50}=2\times\sqrt{25\times2}=2\times5\sqrt{2}=10\sqrt{2}\approx10\sqrt{2}\approx14.14\) (since \(\sqrt{2}\approx1.414\)).

Step2: Order the values

Now we have the approximate values: \(0.5\sqrt{170}\approx6.52\), \(\sqrt{120}\approx10.95\), \(2\sqrt{50}\approx14.14\).
Ordering from least to greatest: \(0.5\sqrt{170}\), \(\sqrt{120}\), \(2\sqrt{50}\) which matches option A.

Answer:

A. \(0.5\sqrt{170}\), \(\sqrt{120}\), \(2\sqrt{50}\)