The square root of h is greater than 10/3 and less than 24/5. How many integer values of h satisfy this condition? I think the first step would be to square each fraction so 100/9 and 576/25 and then see which whole numbers are between. Am I on the right track?

yes

10/3 < √x < 24/5
square it
100/9 ≤ x ≤ 576/25
11.111.. < x < 23.04

so with whole numbers ...
{ 12, 13, ... 22, 23}

count them up

(just realized I used x instead of h, no big deal, eh)

Yes, you are on the right track! To find the number of integer values of h that satisfy the given condition, you need to find the integers between the square roots of 10/3 and 24/5.

As you mentioned, the first step is to square the fractions to get 100/9 and 576/25. Now, you need to find the square root of these fractions:

√(100/9) ≈ 10/3
√(576/25) = 24/5

So, you are looking for the values of h that have square roots greater than 10/3 and less than 24/5. Since square roots are always positive, you can conclude that you are looking for integers between 11 and 23.

The integers that satisfy this condition are 12, 13, 14, ..., 22.

Therefore, there are 22 - 12 + 1 = 11 integer values of h that satisfy the given condition.