Question

estimate the value of \\(\sqrt{11}\\) to the nearest hundredth. drag the number to its approximate location on the number line. 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4

Explanation:

Step1: Identify perfect squares around 11

We know that \(3^2 = 9\) and \(4^2=16\). So \(\sqrt{11}\) is between 3 and 4. Also, \(3.3^2 = 10.89\) and \(3.4^2=11.56\). So \(\sqrt{11}\) is between 3.3 and 3.4.

Step2: Calculate the difference

Let's let \(x = \sqrt{11}\), we know that \(x^2=11\). We know \(3.3^2 = 10.89\), the difference between 11 and 10.89 is \(11 - 10.89=0.11\). The difference between \(3.4^2\) and \(3.3^2\) is \(11.56 - 10.89 = 0.67\).
We can use linear approximation or just test values. Let's try \(3.31^2=3.31\times3.31 = 10.9561\), \(3.32^2=3.32\times3.32 = 11.0224\). Since \(3.31^2 = 10.9561<11\) and \(3.32^2=11.0224>11\), we check the distance. \(11 - 10.9561 = 0.0439\) and \(11.0224 - 11=0.0224\). Since \(0.0224<0.0439\), \(\sqrt{11}\) is closer to 3.32. We can also use a calculator - like approach: \(\sqrt{11}\approx3.3166\), which to the nearest hundredth is 3.32.

Answer:

\(3.32\)