Okay, let's see. The problem says the square root of t is greater than 2 and less than 3.5. I need to find how many integer values of t satisfy this condition. Hmm. So, first, maybe I should write down the inequality as given. 

The square root of t is between 2 and 3.5. So mathematically, that would be 2 < sqrt(t) < 3.5. Right. Now, since sqrt(t) is between those two numbers, I can square all parts of the inequality to find the range of t. 

Squaring the lower bound first: 2 squared is 4. Then the upper bound: 3.5 squared. Let me calculate that. 3.5 times 3.5. 3 times 3 is 9, 3 times 0.5 is 1.5, 0.5 times 3 is another 1.5, and 0.5 times 0.5 is 0.25. Adding those up: 9 + 1.5 + 1.5 + 0.25 = 12.25. So 3.5 squared is 12.25. 

Therefore, the inequality 2 < sqrt(t) < 3.5 becomes 4 < t < 12.25. But t has to be an integer, right? So t must be an integer greater than 4 and less than 12.25. Wait, but 12.25 is not an integer, so the maximum integer t can be is 12. So the integers t can take are from 5 up to 12. Let me check that again. 

If t is greater than 4, the smallest integer t is 5. Then t can be 5, 6, 7, 8, 9, 10, 11, 12. Let me count those. 5,6,7,8,9,10,11,12. That's 8 numbers. Wait, but is 12 included? Because t has to be less than 12.25, so 12 is less than 12.25, so yes, 12 is included. So the integers are 5 through 12 inclusive. Let's count: 12 - 5 +1 = 8. So 8 integers. 

But wait, let me double-check. Let's take t=5. sqrt(5) is approximately 2.236, which is between 2 and 3.5. t=12. sqrt(12) is about 3.464, which is also between 2 and 3.5. So both 5 and 12 are included. So the numbers are 5,6,7,8,9,10,11,12. That's 8 numbers. So the answer should be 8. 

Wait, but let me confirm once more. The original inequality is 2 < sqrt(t) < 3.5. Squaring gives 4 < t < 12.25. Since t must be an integer, the integers are from 5 to 12 inclusive. So how many integers is that? The number of integers from a to b inclusive is b - a + 1. So 12 -5 +1 = 8. Yep. So the answer is 8. 

But just to make sure I didn't make any mistakes in squaring. Let's check with t=5: sqrt(5) ≈2.236, which is greater than 2 and less than 3.5. t=12: sqrt(12)=3.464, still less than 3.5. So both endpoints are correct. 

Is there any integer between 4 and 5? No, 4 is excluded because t must be greater than 4, so the next integer is 5. Similarly, 12.25 is the upper limit, so 12 is included. So everything checks out. 

Therefore, the number of integer values of t that satisfy the condition is 8.
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