What whole number and rational number to the nearest half is square root of 105 between

Answer 1

To find the whole number and the rational number to the nearest half that the square root of 105 falls between, let's first estimate $\sqrt{105}$.

We know that:

• $\sqrt{100}=10$
• $\sqrt{121}=11$

Since $105$ is between $100$ and $121$, it follows that: $$10<\sqrt{105}<11$$

Next, let's approximate $\sqrt{105}$ more closely. Squaring some fractions can help us narrow down the value:

• ${10.2}^2=104.04$
• ${10.3}^2=106.09$

From this, we see that: $$10.2<\sqrt{105}<10.3$$

To find the nearest half, we can consider the halves around $10$:

• $10$
• $10.5$

Since $10.2$ is closer to $10.5$ than to $10$, we round $\sqrt{105}$ to: $$10.5$$

Thus, the square root of $105$ is between the whole number $10$ and the rational number $10.5$ when rounded to the nearest half.