To determine which number lies between [tex]\(\pi\)[/tex] and [tex]\(\sqrt{14}\)[/tex], let's find both values and then compute their average to identify the middle point.
1. Value of [tex]\(\pi\)[/tex]:
[tex]\(\pi\)[/tex] is a well-known mathematical constant, approximately equal to 3.141592653589793.
2. Value of [tex]\(\sqrt{14}\)[/tex]:
The square root of 14 is approximately 3.7416573867739413.
3. Finding the middle value:
To find a number exactly between these two values, we calculate their average. The formula for the average (or the midpoint) of two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is [tex]\(\frac{a + b}{2}\)[/tex].
Plugging in our values:
[tex]\[
\text{Middle value} = \frac{\pi + \sqrt{14}}{2} = \frac{3.141592653589793 + 3.7416573867739413}{2}
\][/tex]
4. Performing the calculation:
Adding the values:
[tex]\[
3.141592653589793 + 3.7416573867739413 = 6.883250040363734
\][/tex]
Dividing by 2:
[tex]\[
\frac{6.883250040363734}{2} = 3.441625020181867
\][/tex]
Therefore, the number that lies between [tex]\(\pi\)[/tex] and [tex]\(\sqrt{14}\)[/tex] is approximately 3.441625020181867.
