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Solved on Jan 16, 2024

Find the best decimal approximation for $\sqrt{62}$ .

STEP 1

Assumptions
1. We need to find the best decimal approximation for the square root of 62.
2. We are given three options: 7.4, 7.5, and 7.9.

STEP 2

To find the best approximation, we can square each of the given options and see which one is closest to 62.

STEP 3

First, we square the option 7.4.
$7.4^2 = (7 + 0.4)^2$

STEP 4

Use the formula $(a + b)^2 = a^2 + 2ab + b^2$ to expand the square of the sum.
$7.4^2 = 7^2 + 2 \cdot 7 \cdot 0.4 + 0.4^2$

STEP 5

Calculate the square of 7 and the other terms.
$7.4^2 = 49 + 2 \cdot 7 \cdot 0.4 + 0.16$

STEP 6

Multiply 2, 7, and 0.4.
$7.4^2 = 49 + 2 \cdot 7 \cdot 0.4 + 0.16 = 49 + 5.6 + 0.16$

STEP 7

Add the terms to find the square of 7.4.
$7.4^2 = 49 + 5.6 + 0.16 = 54.76$

STEP 8

Now, we square the option 7.5.
$7.5^2 = (7 + 0.5)^2$

STEP 9

Use the formula $(a + b)^2 = a^2 + 2ab + b^2$ to expand the square of the sum.
$7.5^2 = 7^2 + 2 \cdot 7 \cdot 0.5 + 0.5^2$

STEP 10

Calculate the square of 7 and the other terms.
$7.5^2 = 49 + 2 \cdot 7 \cdot 0.5 + 0.25$

STEP 11

Multiply 2, 7, and 0.5.
$7.5^2 = 49 + 2 \cdot 7 \cdot 0.5 + 0.25 = 49 + 7 + 0.25$

STEP 12

Add the terms to find the square of 7.5.
$7.5^2 = 49 + 7 + 0.25 = 56.25$

STEP 13

Finally, we square the option 7.9.
$7.9^2 = (7 + 0.9)^2$

STEP 14

Use the formula $(a + b)^2 = a^2 + 2ab + b^2$ to expand the square of the sum.
$7.9^2 = 7^2 + 2 \cdot 7 \cdot 0.9 + 0.9^2$

STEP 15

Calculate the square of 7 and the other terms.
$7.9^2 = 49 + 2 \cdot 7 \cdot 0.9 + 0.81$

STEP 16

Multiply 2, 7, and 0.9.
$7.9^2 = 49 + 2 \cdot 7 \cdot 0.9 + 0.81 = 49 + 12.6 + 0.81$

STEP 17

Add the terms to find the square of 7.9.
$7.9^2 = 49 + 12.6 + 0.81 = 62.41$

STEP 18

Now that we have the squares of all the options, we compare them to 62 to see which is closest.

STEP 19

Compare the square of 7.4 to 62.
$|62 - 54.76| = 7.24$

STEP 20

Compare the square of 7.5 to 62.
$|62 - 56.25| = 5.75$

STEP 21

Compare the square of 7.9 to 62.
$|62 - 62.41| = 0.41$

STEP 22

The best approximation is the one with the smallest difference from 62.

STEP 23

Since $0.41 < 5.75 < 7.24$ , the square of 7.9 is closest to 62.
The best decimal approximation for the square root of $62$ is 7.9.

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