Question

select the correct answer.

on the number line, which point is closest to ??

Explanation:

Determine the value of the square root

We need to find which point on the number line is closest to \(\sqrt{10}\).
Using the Estimating Square Roots concept, we know that:

$$ 3^2 = 9 \quad \text{and} \quad 4^2 = 16 $$

Since \(9 < 10 < 16\), it follows that \(3 < \sqrt{10} < 4\).

Refine the approximation

To find a closer decimal approximation, we test values between \(3.1\) and \(3.2\):

$$ 3.1^2 = 9.61 $$

$$ 3.2^2 = 10.24 $$

Since \(10\) is between \(9.61\) and \(10.24\), the value of \(\sqrt{10}\) lies between \(3.1\) and \(3.2\).

Calculate the exact distance to find the closest point

Let's find which tenth value is closer to \(\sqrt{10}\) by comparing the differences:

$$ 10 - 3.1^2 = 10 - 9.61 = 0.39 $$

$$ 3.2^2 - 10 = 10.24 - 10 = 0.24 $$

Since \(0.24 < 0.39\), the value \(10\) is closer to \(10.24\) than to \(9.61\). Therefore, \(\sqrt{10}\) is closer to \(3.2\) than to \(3.1\).
Using a calculator for high precision:

$$ \sqrt{10} \approx 3.162 $$

Analyze the number line intervals

The number line has major tick marks labeled at \(3.1\) and \(3.2\).
Let's count the subdivisions between \(3.1\) and \(3.2\):
There are exactly \(10\) equal intervals between \(3.1\) and \(3.2\).
Each small subdivision represents:

$$ \frac{3.2 - 3.1}{10} = 0.01 $$

Thus, the tick marks starting from \(3.1\) are:
\(3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16, 3.17, 3.18, 3.19, 3.20\).

Identify the positions of the points

Let's locate each point on this scale:

• Point A is at \(3.11\).
• Point B is at \(3.13\).
• Point C is at \(3.15\).
• Point D is at \(3.19\).

We compare these positions to our target value \(\sqrt{10} \approx 3.162\):

• Distance to A: \(|3.162 - 3.11| = 0.052\)
• Distance to B: \(|3.162 - 3.13| = 0.032\)
• Distance to C: \(|3.162 - 3.15| = 0.012\)
• Distance to D: \(|3.162 - 3.19| = 0.028\)

The point with the smallest distance to \(3.162\) is Point C.

Answer:

• A
• B
• C (Correct answer)
• D