Select the correct answer.
Between which values does [tex]$\sqrt{111}$[/tex] lie?
A. 7 and 8
B. 8 and 9
C. 9 and 10
D. 10 and 11
E. 11 and 12
Answer (1)
We need to find two consecutive integers between which 111 lies.
Calculate the squares of the given integers: 7 2 = 49 , 8 2 = 64 , 9 2 = 81 , 1 0 2 = 100 , 1 1 2 = 121 , 1 2 2 = 144 .
Determine that 1 0 2 < 111 < 1 1 2 , so 10 < 111 < 11 .
Therefore, 111 lies between 10 an d 11 .
Explanation
Problem Analysis We need to find two consecutive integers n and n + 1 such that n < 111 < n + 1 . Equivalently, we need to find n such that n 2 < 111 < ( n + 1 ) 2 .
Solution Strategy Let's calculate the squares of the integers in the given options to determine which two consecutive squares bound the number 111.
Calculating Squares We have: 7 2 = 49 8 2 = 64 9 2 = 81 1 0 2 = 100 1 1 2 = 121 1 2 2 = 144
Determining the Interval Now, we check which two consecutive squares bound 111. We see that 100 < 111 < 121 , which means 1 0 2 < 111 < 1 1 2 . Therefore, 10 < 111 < 11 .
Final Answer Thus, 111 lies between 10 and 11.
Examples
Estimating square roots is useful in various real-life situations. For instance, if you're planning a garden and want to fence off a square area of approximately 111 square meters, knowing that 111 is between 10 and 11 tells you that each side of the square should be a little over 10 meters. This helps you buy the right amount of fencing material without needing a precise calculation.
