Math / Numbers & Operations

QuestionWhich of the following statements is true? $\sqrt{105}$ is between - 9 and 10 , but closer to 10. 9 and 10 , but closer to 11. 10 and 11 , but closer to 10 . 10 and 11 , but closer to 11

Studdy Solution

STEP 1

1. We need to determine the approximate value of $\sqrt{105}$ .
2. We will compare $\sqrt{105}$ to integer values to determine which range it falls into and which integer it is closer to.

STEP 2

1. Estimate the approximate value of $\sqrt{105}$ .
2. Determine the range in which $\sqrt{105}$ falls.
3. Determine which integer $\sqrt{105}$ is closer to.

STEP 3

Estimate the approximate value of $\sqrt{105}$ by identifying perfect squares around $105$ .
The perfect squares around $105$ are $100$ and $121$ .

STEP 4

Calculate the square roots of these perfect squares:
$\sqrt{100} = 10$ $\sqrt{121} = 11$
Since $105$ is between $100$ and $121$ , $\sqrt{105}$ is between $10$ and $11$ .

STEP 5

Determine which integer $\sqrt{105}$ is closer to by estimating further:
Calculate the square of $10.5$ to see if $\sqrt{105}$ is closer to $10$ or $11$ .
$10.5^2 = 110.25$
Since $105$ is less than $110.25$ , $\sqrt{105}$ is closer to $10$ than $11$ .
The correct statement is:
$\boxed{\text{10 and 11, but closer to 10}}$

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