Question Find the integer between $\frac{11 \pi}{5}$ and $\sqrt{58}$ .

Studdy Solution

STEP 1

Assumptions
1. We are looking for an integer that lies between $\frac{11 \pi}{5}$ and $\sqrt{58}$ .
2. We will approximate the values of $\pi$ and $\sqrt{58}$ to sufficient accuracy to determine the integer between them.

STEP 2

First, we need to approximate the value of $\pi$ . We know that $\pi \approx 3.14159$ .

STEP 3

Now, we calculate the value of $\frac{11 \pi}{5}$ using the approximation of $\pi$ .
$\frac{11 \pi}{5} = \frac{11 \times 3.14159}{5}$

STEP 4

Perform the multiplication and division to find the approximate value.
$\frac{11 \times 3.14159}{5} \approx \frac{11 \times 3.14}{5}$
$\frac{11 \times 3.14}{5} \approx \frac{34.54}{5}$

STEP 5

Calculate the approximate value of $\frac{11 \pi}{5}$ .
$\frac{34.54}{5} \approx 6.908$

STEP 6

Next, we need to approximate the value of $\sqrt{58}$ . We know that $\sqrt{49} = 7$ and $\sqrt{64} = 8$ , so $\sqrt{58}$ must be between 7 and 8.

STEP 7

To get a more precise approximation, we can use the fact that $58$ is closer to $49$ than to $64$ , so $\sqrt{58}$ should be closer to $7$ than to $8$ .

STEP 8

We can estimate $\sqrt{58}$ to be approximately $7.6$ since it is closer to $7$ but not as close to $8$ .

STEP 9

Now we have the approximate values of both $\frac{11 \pi}{5}$ and $\sqrt{58}$ .
$\frac{11 \pi}{5} \approx 6.908$ $\sqrt{58} \approx 7.6$

STEP 10

We can see that the integer 7 is between the two approximate values.
$6.908 < 7 < 7.6$

STEP 11

Therefore, the integer that lies between $\frac{11 \pi}{5}$ and $\sqrt{58}$ is 7.
The integer is 7.

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