Solved on Dec 10, 2023

Find the nearest integer to $\sqrt{59}$ .

STEP 1

Assumptions
1. We are looking for the nearest integer approximation to the square root of 59.
2. We will use estimation and possibly the method of successive approximations (also known as the Babylonian method or Newton's method) if necessary.

STEP 2

Identify perfect squares near 59 to establish a starting point for our estimation.

STEP 3

Recognize that $7^2 = 49$ and $8^2 = 64$ . Since 59 is between 49 and 64, the square root of 59 will be between 7 and 8.

STEP 4

Since 59 is closer to 64 than it is to 49, we can expect that $\sqrt{59}$ will be closer to 8 than to 7.

STEP 5

To refine our estimate, consider the middle point between 7 and 8, which is 7.5. Calculate the square of 7.5 to see if it is closer to 59.
$7.5^2 = 56.25$

STEP 6

Since $56.25 < 59$ , we know that $\sqrt{59}$ is greater than 7.5 but less than 8.

STEP 7

Now, consider a number slightly larger than 7.5, for example, 7.6, and calculate its square.
$7.6^2 = 57.76$

STEP 8

Since $57.76 < 59$ , we can further narrow down our approximation to between 7.6 and 8.

STEP 9

Next, consider a number slightly larger than 7.6, for example, 7.7, and calculate its square.
$7.7^2 = 59.29$

STEP 10

Since $59.29 > 59$ , we know that $\sqrt{59}$ is less than 7.7 but greater than 7.6.

STEP 11

Given that 7.6 squared is less than 59 and 7.7 squared is more than 59, and our goal is to find the nearest integer, we can conclude that the nearest integer to $\sqrt{59}$ is 8, because $\sqrt{59}$ is closer to 8 than to 7.
The approximate solution to $x=\sqrt{59}$ to the nearest integer is 8.

Was this helpful?

Find the nearest integer to $\sqrt{59}$ .

STEP 1

Assumptions
1. We are looking for the nearest integer approximation to the square root of 59.
2. We will use estimation and possibly the method of successive approximations (also known as the Babylonian method or Newton's method) if necessary.

STEP 2

Identify perfect squares near 59 to establish a starting point for our estimation.

STEP 3

Recognize that $7^2 = 49$ and $8^2 = 64$ . Since 59 is between 49 and 64, the square root of 59 will be between 7 and 8.

STEP 4

Since 59 is closer to 64 than it is to 49, we can expect that $\sqrt{59}$ will be closer to 8 than to 7.

STEP 5

To refine our estimate, consider the middle point between 7 and 8, which is 7.5. Calculate the square of 7.5 to see if it is closer to 59.
$7.5^2 = 56.25$

STEP 6

Since $56.25 < 59$ , we know that $\sqrt{59}$ is greater than 7.5 but less than 8.

STEP 7

Now, consider a number slightly larger than 7.5, for example, 7.6, and calculate its square.
$7.6^2 = 57.76$

STEP 8

Since $57.76 < 59$ , we can further narrow down our approximation to between 7.6 and 8.

STEP 9

Next, consider a number slightly larger than 7.6, for example, 7.7, and calculate its square.
$7.7^2 = 59.29$

STEP 10

Since $59.29 > 59$ , we know that $\sqrt{59}$ is less than 7.7 but greater than 7.6.

STEP 11

Given that 7.6 squared is less than 59 and 7.7 squared is more than 59, and our goal is to find the nearest integer, we can conclude that the nearest integer to $\sqrt{59}$ is 8, because $\sqrt{59}$ is closer to 8 than to 7.
The approximate solution to $x=\sqrt{59}$ to the nearest integer is 8.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the nearest integer to 59\sqrt{59}59​.

STEP 1

Assumptions1. We are looking for the nearest integer approximation to the square root of 59.2. We will use estimation and possibly the method of successive approximations (also known as the Babylonian method or Newton's method) if necessary.

STEP 2

Identify perfect squares near 59 to establish a starting point for our estimation.

STEP 3

Recognize that 72=497^2 = 4972=49 and 82=648^2 = 6482=64. Since 59 is between 49 and 64, the square root of 59 will be between 7 and 8.

STEP 4

Since 59 is closer to 64 than it is to 49, we can expect that 59\sqrt{59}59​ will be closer to 8 than to 7.

STEP 5

To refine our estimate, consider the middle point between 7 and 8, which is 7.5. Calculate the square of 7.5 to see if it is closer to 59.7.52=56.257.5^2 = 56.257.52=56.25

STEP 6

Since 56.25<5956.25 < 5956.25<59, we know that 59\sqrt{59}59​ is greater than 7.5 but less than 8.

STEP 7

Now, consider a number slightly larger than 7.5, for example, 7.6, and calculate its square.7.62=57.767.6^2 = 57.767.62=57.76

STEP 8

Since 57.76<5957.76 < 5957.76<59, we can further narrow down our approximation to between 7.6 and 8.

STEP 9

Next, consider a number slightly larger than 7.6, for example, 7.7, and calculate its square.7.72=59.297.7^2 = 59.297.72=59.29

STEP 10

Since 59.29>5959.29 > 5959.29>59, we know that 59\sqrt{59}59​ is less than 7.7 but greater than 7.6.

STEP 11

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the nearest integer to $\sqrt{59}$ .

Assumptions
1. We are looking for the nearest integer approximation to the square root of 59.
2. We will use estimation and possibly the method of successive approximations (also known as the Babylonian method or Newton's method) if necessary.

Recognize that $7^2 = 49$ and $8^2 = 64$ . Since 59 is between 49 and 64, the square root of 59 will be between 7 and 8.

Since 59 is closer to 64 than it is to 49, we can expect that $\sqrt{59}$ will be closer to 8 than to 7.

To refine our estimate, consider the middle point between 7 and 8, which is 7.5. Calculate the square of 7.5 to see if it is closer to 59.
$7.5^2 = 56.25$

Since $56.25 < 59$ , we know that $\sqrt{59}$ is greater than 7.5 but less than 8.

Now, consider a number slightly larger than 7.5, for example, 7.6, and calculate its square.
$7.6^2 = 57.76$

Since $57.76 < 59$ , we can further narrow down our approximation to between 7.6 and 8.

Next, consider a number slightly larger than 7.6, for example, 7.7, and calculate its square.
$7.7^2 = 59.29$

Since $59.29 > 59$ , we know that $\sqrt{59}$ is less than 7.7 but greater than 7.6.