Using a number line, what whole number and rational number to the nearest half is 97−−√ between?(1 point)
On a number line, we can find the whole number and rational number to the nearest half that is closest to $\sqrt{97}$.
First, we find the two perfect squares on either side of 97: $9^2=81$ and $10^2=100$.
Since $9^2<97<10^2$, we know that $\sqrt{97}$ is between 9 and 10.
Next, we find the midpoint between 9 and 10, which is $\frac{9+10}{2}=\frac{19}{2}=\boxed{9.5}$. Therefore, the whole number and rational number to the nearest half that is closest to $\sqrt{97}$ is 9.5.
using rational approximations, what is an inequality statement comparing 21−−√ and 17−−√? Enter <, >, or =.(1 point)
To compare $\sqrt{21}$ and $\sqrt{17}$ using rational approximations, we can find the integers closest to each square root.
For $\sqrt{21}$:
$4^2 = 16$ and $5^2 = 25$
Since $16 < 21 < 25$, we know that $\sqrt{21}$ is between 4 and 5.
For $\sqrt{17}$:
$4^2 = 16$ and $5^2 = 25$
Since $16 < 17 < 25$, we know that $\sqrt{17}$ is also between 4 and 5.
Therefore, $\sqrt{21}$ and $\sqrt{17}$ are both between 4 and 5.
Inequality statement: $\sqrt{21}$ = $\sqrt{17}$.
$\boxed{=}$
Using rational approximations, what is an inequality statement comparing π and 8–√? Enter <, >, or =.(1 point)
To compare $\pi$ and $\sqrt{8}$ using rational approximations, we can use the fact that $\pi$ is an irrational number approximately equal to 3.14.
For $\sqrt{8}$:
$2^2 = 4$ and $3^2 = 9$
Since $4 < 8 < 9$, we know that $\sqrt{8}$ is between 2 and 3.
Therefore, $\pi$ is approximately 3.14, which is larger than $\sqrt{8}$.
Inequality statement: $\pi > \sqrt{8}$.
$\boxed{>}$
To find the whole number and rational number nearest to √97, let's start by calculating the value of √97.
Using a calculator, we find that √97 is approximately 9.8488578.
Now, let's find the whole number and rational number closest to √97:
On a number line, we can plot the value of √97 between the whole numbers 9 and 10, since 9 is less than √97 and 10 is greater than √97.
Now, to find the rational number to the nearest half, we can divide the interval between 9 and 10 into halves. The midpoint of this interval will give us our rational number.
The midpoint between 9 and 10 is 9.5.
Therefore, the whole number nearest to √97 is 10, and the rational number to the nearest half is 9.5.
To find the whole number and rational number to the nearest half that 97−−√ lies between using a number line, we can follow these steps:
1. Start by identifying the whole numbers that are closest to 97−−√.
The whole number closest to 97−√ is the largest whole number whose square is less than or equal to 97. In this case, since 10^2 = 100, and 100 is greater than 97, the largest whole number whose square is less than 97 is 9.
The whole number closest to 97+√ is the smallest whole number whose square is greater than 97. In this case, since 11^2 = 121, and 121 is greater than 97, the smallest whole number whose square is greater than 97 is 11.
2. Determine the rational numbers closest to 97−−√ that are halfway between the two whole numbers identified in step 1.
The rational number halfway between 9 and 11 can be found by taking their average. Add 9 and 11 together and divide by 2:
(9 + 11) / 2 = 20 / 2 = 10.
Therefore, the rational number halfway between 9 and 11 is 10.
Hence, the whole number and rational number to the nearest half that 97−−√ lies between on a number line are 9 and 10, respectively.