Show an example of a rational number that is between 10−−√ and 11−−√.
a 10/3
b 16/5
c 3.1
d 3.4
c) 3.1
Sure?
I apologize for the confusion in my previous response. My answer was incorrect. The correct answer is:
d) 3.4
Is that correct?
I apologize again for the confusion. My previous response was incorrect.
The correct answer is:
b) 16/5
To show that this fraction is between √10 and √11, we can calculate their approximate decimal values:
√10 ≈ 3.162
√11 ≈ 3.317
16/5 ≈ 3.2
Since 3.162 < 3.2 < 3.317, we can conclude that 16/5 is indeed between √10 and √11.
To determine which of the given options is a rational number between √10 and √11, we can examine each option.
a) Option a: 10/3 is a rational number, but it is greater than both √10 and √11, so it is not between them.
b) Option b: 16/5 is also a rational number, but it is greater than both √10 and √11, so it is not between them either.
c) Option c: 3.1 is a decimal number, and it is less than √11. However, it is greater than √10, so it is a valid option.
d) Option d: 3.4 is another decimal number, and it is also less than √11. However, it is greater than √10, so it is also a valid option.
Therefore, the rational numbers that are between √10 and √11 are options c) 3.1 and d) 3.4.
To determine whether a rational number is between √10 and √11, we need to compare their decimal approximations.
Let's find the decimal approximations of √10 and √11.
√10 is approximately 3.16227766017 (rounded to 11 decimal places)
√11 is approximately 3.31662479036 (rounded to 11 decimal places)
Now, let's analyze the given options:
a) 10/3 = 3.33333333333
b) 16/5 = 3.2
c) 3.1
d) 3.4
Among the options, option c (3.1) does not fall between the decimal approximations of √10 and √11.
However, options a (10/3) and b (16/5) are both greater than √10 but less than √11, making them the rational numbers that lie between √10 and √11.
So, the correct answer is a) 10/3.