Using a number line, what whole number and rational number to the nearest half is √105 between?
A. Between 9.5 and 10
B. Between 10.5 and 11
C. between 10 and 10.5
D. between 11 and 11.5
First, we need to find the whole number that √105 is between. The approximate value of √105 is 10.25.
Next, we need to find the nearest half to 10.25. The nearest half is 10 because 10 is closer to 10.25 than 10.5 or any other half.
So, √105 is between the whole numbers 10 and 11.
Therefore, the whole number and rational number to the nearest half that √105 is between is D. between 11 and 11.5.
how did you get √105 s between 10 and 11 but come up with 11 and 11.5?
Apologies for the confusion in my previous response. To clarify:
√105 is between 10 and 11 on the number line.
To find the rational number to the nearest half, we need to determine whether √105 is closer to 10 or 11.
To determine this, we can calculate the midpoints between 10 and 10.5, and between 10.5 and 11:
10 + 10.25 = 20.25 / 2 = 10.125
10.5 + 11 = 21.5 / 2 = 10.75
Since the value of √105 is closer to 10.125 than to 10.75, the rational number to the nearest half for √105 is between 10 and 10.5.
Therefore, the correct answer is C. between 10 and 10.5.
To determine which whole number and rational number to the nearest half √105 is between, we can calculate the square root of 105. The square root of 105 is approximately 10.247.
Now, we'll consider the given options:
A. Between 9.5 and 10: The whole number 10 is less than √105.
B. Between 10.5 and 11: The whole number 11 is greater than √105.
C. Between 10 and 10.5: The whole number 10 is less than √105.
D. Between 11 and 11.5: The whole number 11 is greater than √105.
Therefore, √105 is between the whole numbers 10 and 11. But we also need to find the rational number to the nearest half.
To determine the rational number to the nearest half, we can round the decimal value 10.247 to the nearest half. The nearest half for 10.247 is 10.5.
So, the answer is:
C. Between 10 and 10.5
To determine the whole number and rational number to the nearest half that is closest to √105, we can start by finding the approximate value of √105.
The value of √105 can be estimated by finding the square root of the closest perfect square values on either side of 105. In this case, the perfect square values that are closest to 105 are 100 (10^2) and 121 (11^2).
By comparing the distance from √105 to the square roots of 100 and 121, we can determine which number line segment √105 falls within.
√100 = 10
√121 = 11
The value of √105 lies between 10 and 11.
Now let's consider the number line intervals provided as answer choices:
A. Between 9.5 and 10
B. Between 10.5 and 11
C. between 10 and 10.5
D. between 11 and 11.5
Since √105 is between 10 and 11, we can eliminate options A and D.
Next, let's consider option B, which states that √105 is between 10.5 and 11. This interval includes 10.5, but not 10. Therefore, option B is a possible answer.
Finally, let's consider option C, which states that √105 is between 10 and 10.5. This interval includes 10, but not 10.5. Therefore, option C is the other possible answer.
In conclusion, the whole number and rational number to the nearest half that √105 is between are both options B (Between 10.5 and 11) and C (Between 10 and 10.5).