What whole number and rational number to the nearest half is square root of 105 between

1 answer

To find the whole number and the rational number to the nearest half that the square root of 105 falls between, let's first estimate 105.

We know that:

  • 100=10
  • 121=11

Since 105 is between 100 and 121, it follows that: 10<105<11

Next, let's approximate 105 more closely. Squaring some fractions can help us narrow down the value:

  • 10.22=104.04
  • 10.32=106.09

From this, we see that: 10.2<105<10.3

To find the nearest half, we can consider the halves around 10:

  • 10
  • 10.5

Since 10.2 is closer to 10.5 than to 10, we round 105 to: 10.5

Thus, the square root of 105 is between the whole number 10 and the rational number 10.5 when rounded to the nearest half.