QuestionCount the integers between $\sqrt{99}$ and $\sqrt{297}$ .

Studdy Solution

STEP 1

Assumptions1. We are looking for integers that lie between $\sqrt{99}$ and $\sqrt{297}$ . . The integers are inclusive, meaning they can also be equal to $\sqrt{99}$ or $\sqrt{297}$ if those values are integers.

STEP 2

First, we need to find the square root of99.
$\sqrt{99}$

STEP 3

Calculate the square root of99.
$\sqrt{99} \approx9.95$

STEP 4

Since we are looking for integers, we need to round up the square root of99 to the nearest integer. This is because we are looking for integers that are greater than or equal to $\sqrt{99}$ .
$\lceil\sqrt{99}\rceil =10$

STEP 5

Next, we need to find the square root of297.
$\sqrt{297}$

STEP 6

Calculate the square root of297.
$\sqrt{297} \approx17.23$

STEP 7

Since we are looking for integers, we need to round down the square root of297 to the nearest integer. This is because we are looking for integers that are less than or equal to $\sqrt{297}$ .
$\lfloor\sqrt{297}\rfloor =17$

STEP 8

Now that we have the range of integers, we can find the number of integers that lie between10 and17, inclusive.
This can be found by subtracting the lower bound from the upper bound and adding1 (since both bounds are inclusive).
$Number\, of\, integers = Upper\, bound - Lower\, bound +1$

STEP 9

Plug in the values for the upper bound and the lower bound to calculate the number of integers.
$Number\, of\, integers =17 - +$

STEP 10

Calculate the number of integers.
$Number\, of\, integers =17 -10 + =8$ There are8 integers that lie between $\sqrt{99}$ and $\sqrt{297}$ .

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