Using a number line, what whole number and rational number to the nearest half is 97−−√ between?
To find the whole number and rational number to the nearest half that is between $\sqrt{97}$, we can look for the two integers between which $\sqrt{97}$ lies.
First, we note that $10^2=100$ is larger than $97$. Since $\sqrt{100}=10$, $\sqrt{97}$ is a little less than 10.
We can compute by hand or with a calculator that $9^2=81$ and $10^2=100$, so $\sqrt{81}<\sqrt{97}<\sqrt{100}$.
Therefore, the whole number that $\sqrt{97}$ is between is 9.
Next, we can estimate the fractional part by dividing up the interval between 9 and 10 into halves. Since $\sqrt{97}$ is closer to 10, the fractional part will be closer to one than to zero. Therefore, $\sqrt{97}$ is closer to $9\dfrac{1}{2}$ than to $9\dfrac{0}{2}$.
So, the whole number and rational number to the nearest half that $\sqrt{97}$ is between are $\boxed{9\text{ and }\dfrac{9}{2}}$.
Using a number line, what whole number and rational number to the nearest half is 97−−√ between?
To find the whole number and rational number to the nearest half that $\sqrt{97}$ is between, we can use a number line.
First, we know that $9^2 = 81$ and $10^2 = 100$. Therefore, $\sqrt{97}$ must be between 9 and 10.
Next, we can divide the interval between 9 and 10 into halves. The midpoint of this interval is $9\frac{1}{2}$, which is exactly halfway between 9 and 10.
Since $\sqrt{97}$ is closer to 10 than to 9, the nearest whole number is 10.
Looking at the halves, we see that $\sqrt{97}$ is closer to $9\frac{1}{2}$ than to $9\frac{0}{2}$.
Therefore, the whole number and rational number to the nearest half that $\sqrt{97}$ is between are $\boxed{10\text{ and } 9\frac{1}{2}}$.
Using rational approximations, what is an inequality statement comparing 21−−√ and 17−−√? Enter <, >, or =.
To compare $\sqrt{21}$ and $\sqrt{17}$, we can use rational approximations.
For $\sqrt{21}$, we can estimate that $21$ is between $16$ and $25$, and $\sqrt{16}=4$ and $\sqrt{25}=5$. Therefore, $4< \sqrt{21}< 5$.
For $\sqrt{17}$, we can estimate that $17$ is between $16$ and $18$, and $\sqrt{16}=4$ and $\sqrt{18}\approx4.24$. Therefore, $4< \sqrt{17}<4.24$.
Comparing the two values, we have $4< \sqrt{17}< \sqrt{21}<5$.
Thus, the inequality statement comparing $\sqrt{21}$ and $\sqrt{17}$ is $\sqrt{17}<\sqrt{21}$.
Using a number line, what is the approximate value of 19−−√?(1 point)
To approximate the value of $\sqrt{19}$ using a number line, we can start by identifying the two perfect square numbers that $\sqrt{19}$ falls between.
Since $4^2=16$ and $5^2=25$, we know that $\sqrt{19}$ is between 4 and 5.
To get a more precise estimate, we can divide the interval between 4 and 5 into smaller parts. One way to do this is by halves. So, we can divide the interval into 2 equal parts:
$$4 \quad \frac{4+5}{2} \quad 5.$$
Calculating the average of 4 and 5, we get $\frac{4+5}{2} = \frac{9}{2} = 4.5$.
Therefore, the approximate value of $\sqrt{19}$ is $\boxed{4.5}$.
Using a number line, what is the approximate value of 19−−√?(1 point)
Responses
4.13
4.5
4.38
4.25
To approximate the value of $\sqrt{19}$ using a number line, we can start by identifying the two perfect square numbers that $\sqrt{19}$ falls between.
Since $4^2=16$ and $5^2=25$, we know that $\sqrt{19}$ is between 4 and 5.
To get a more precise estimate, we can divide the interval between 4 and 5 into smaller parts. One reasonable choice is to divide the interval into tenths:
$$4.0 \quad 4.1 \quad 4.2 \quad 4.3 \quad 4.4 \quad 4.5 \quad 4.6 \quad 4.7 \quad 4.8 \quad 4.9 \quad 5.0.$$
From this division, we see that $\sqrt{19}$ is closer to 4.3 than 4.2, so it is reasonable to estimate the value as approximately $\boxed{4.3}$.