Math / Numbers & Operations

QuestionCompare $\sqrt{42}$ and $\sqrt{58}$ plotted on the number line. What is the approximate difference in tenths between the two values? $\sqrt{58}$ is about $\square$ greater than $\sqrt{42}$ .

Studdy Solution

STEP 1

1. We are comparing the square roots of two numbers: $\sqrt{42}$ and $\sqrt{58}$ .
2. We need to find the approximate difference between these two square roots in tenths.
3. The difference is calculated as $\sqrt{58} - \sqrt{42}$ .

STEP 2

1. Estimate $\sqrt{42}$ .
2. Estimate $\sqrt{58}$ .
3. Calculate the approximate difference.

STEP 3

Estimate $\sqrt{42}$ by finding two perfect squares it falls between.
The perfect squares around $42$ are $36$ and $49$ , so:
$\sqrt{36} = 6 \quad \text{and} \quad \sqrt{49} = 7$
Since $42$ is closer to $36$ , $\sqrt{42}$ is slightly greater than $6$ .
Estimate: $\sqrt{42} \approx 6.5$

STEP 4

Estimate $\sqrt{58}$ by finding two perfect squares it falls between.
The perfect squares around $58$ are $49$ and $64$ , so:
$\sqrt{49} = 7 \quad \text{and} \quad \sqrt{64} = 8$
Since $58$ is closer to $64$ , $\sqrt{58}$ is slightly less than $8$ .
Estimate: $\sqrt{58} \approx 7.6$

STEP 5

Calculate the approximate difference between $\sqrt{58}$ and $\sqrt{42}$ .
$\sqrt{58} - \sqrt{42} \approx 7.6 - 6.5 = 1.1$
The approximate difference in tenths between $\sqrt{58}$ and $\sqrt{42}$ is:
$\boxed{1.1}$

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