Using a number line, what whole number and rational number to the nearest half is √40 between?
√40 is between __ and __.
To find the whole number and rational number to the nearest half that is between √40, we can start by finding the approximate value of √40.
First, we can find the square root of 40:
√40 ≈ 6.324555320336759
Since we are looking for the nearest half value, we can round 6.324555320336759 to the nearest half, which is 6.5.
This tells us that √40 is between 6 and 7 on the number line.
Therefore, √40 is between 6 and 7 as a whole number and between 6.5 and 7 as a rational number to the nearest half.
To determine the whole number and rational number to the nearest half that √40 is between, let's find the perfect squares that √40 lies between.
The perfect square that is smaller than 40 is 36 (6^2), and the perfect square that is larger than 40 is 49 (7^2).
Therefore, √40 lies between √36 and √49.
Calculating the values:
√36 ≈ 6
√49 ≈ 7
So, √40 is between 6 and 7.
To find the rational number to the nearest half, we can take the average of 6 and 7.
(6 + 7) / 2 = 13 / 2 = 6.5
Therefore, √40 is between 6 and 7, and to the nearest half, it is 6.5.
To find the whole number and rational number to the nearest half that √40 is between, we will use a number line.
First, let's determine the whole number that √40 is between. To do this, let's find the perfect squares that surround 40. The perfect square numbers that come before and after 40 are 36 (6^2) and 49 (7^2), respectively.
On the number line, we can place these perfect square numbers:
... 36 ---- √40 ---- 49 ...
Next, we will determine the rational number to the nearest half that √40 is between. We can divide the difference between the perfect squares into halves and find the corresponding points on the number line.
The difference between 49 and 36 is 49 - 36 = 13. Half of 13 is 13/2 = 6.5.
Adding and subtracting 6.5 from the perfect square numbers, we get:
... 36 ---- √40 - 6.5 ---- √40 + 6.5 ---- 49 ...
So, √40 is between √40 - 6.5 and √40 + 6.5.
To get the whole number and rational number to the nearest half, we can simplify the expressions:
√40 - 6.5 ≈ 6 - 6.5 = -0.5 (the nearest whole number to √40)
√40 + 6.5 ≈ 6 + 6.5 = 12.5 (the nearest whole number to √40)
Therefore, √40 is between -0.5 (or -1/2) and 12.5 (or 25/2) on the number line.