Using a number line, what whole number and rational number to the nearest half is 105−−−√ between?
a between 10.5 and 11
b between 10 and 10.5
c between 9.5 and 10
d between 11 and 11.5
To determine the whole number and rational number to the nearest half that is between $10$ and $11$, we need to find the two half numbers that are closest to $\sqrt{105}$. We can estimate that $\sqrt{100}=10$, so $\sqrt{105}$ is slightly greater than $10$. Since $10$ is $20$ half units away from $0$ and $11$ is $22$ half units away from $0$, we know that $\sqrt{105}$ must be closer to $10$ than to $11$. Therefore, the whole number and rational number to the nearest half that $\sqrt{105}$ is between is $\boxed{\text{(a) between 10.5 and 11}}$.
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Apologies for the previous incorrect response. Let's try again:
To determine the whole number and rational number to the nearest half that is between $\sqrt{105}$, we can use estimation. We know that $10^2 = 100$ and $11^2 = 121$, so $\sqrt{105}$ must be somewhere between $10$ and $11$.
To estimate further, we can find the midpoint between $10$ and $11$, which is $10.5$. By comparing this midpoint to $\sqrt{105}$, we can see that $\sqrt{105}$ is closer to $10$ than to $11$.
Therefore, the whole number and rational number to the nearest half that $\sqrt{105}$ is between is $\boxed{\text{(a) between 10.5 and 11}}$.
To find the whole number and rational number to the nearest half that 105−−−√ is between, we can first estimate the value of 105−−−√.
Since the square root of 105 is approximately 10.25, we know that 105−−−√ is slightly greater than 10.
Now let's consider the options:
a) between 10.5 and 11: Since 105−−−√ is slightly greater than 10, it is not between 10.5 and 11.
b) between 10 and 10.5: Since 105−−−√ is slightly greater than 10, it is between 10 and 10.5.
c) between 9.5 and 10: Since 105−−−√ is greater than 10, it is not between 9.5 and 10.
d) between 11 and 11.5: Since 105−−−√ is less than 11, it is not between 11 and 11.5.
Therefore, the correct answer is b) between 10 and 10.5.