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We are given the following functions to solve the problem:

1. The bacteria count \( N(T) \), which depends on the temperature \( T \):
   \[
   N(T) = 3T^2 - 24T + 479
   \]
   where \( 2 \leq T \leq 14 \).

2. The temperature function \( T(t) \), which depends on time \( t \) (in hours):
   \[
   T(t) = 5t + 4
   \]
   where \( 0 \leq t \leq 3 \).

We are asked to find the composite function \( N(T(t)) \), which expresses the bacteria count in terms of time \( t \), and then solve for the time when the bacterial count reaches 1928. 

### Step 1: Form the composite function \( N(T(t)) \)
Substitute \( T(t) = 5t + 4 \) into the function for \( N(T) \):
\[
N(T(t)) = 3(5t + 4)^2 - 24(5t + 4) + 479
\]

### Step 2: Solve for when \( N(T(t)) = 1928 \)
We will set \( N(T(t)) = 1928 \) and solve for \( t \).

Let me first calculate and solve this step.The solutions for \( t \) are \( t = -4 \) and \( t = 4 \). However, since the problem states that time \( t \) is between 0 and 3 (i.e., \( 0 \leq t \leq 3 \)), we discard \( t = -4 \) as it is not within the valid time range.

Therefore, the time when the bacterial count reaches 1928 is approximately \( t = 4 \). However, this solution falls outside the specified time range, so it seems no valid solution exists for the range \( 0 \leq t \leq 3 \).

Would you like me to check any further details or calculations, or help with another problem?

Here are five related questions to expand your understanding:
1. How is a composite function formed in general?
2. What is the domain of a composite function, and how does it relate to the individual functions?
3. Why might a problem have no valid solution within a certain range?
4. How can quadratic equations yield extraneous solutions in context-based problems?
5. How does rounding affect the accuracy of solutions in applied mathematics?

**Tip:** Always check the domain restrictions carefully when solving composite functions.

Find the composite function N(T(t)) and use this equation to find the time when the bacterial count reaches 1928. Round your answer to the nearest whole number.

Learn how to solve a composite function problem involving bacterial growth in food and quadratic equations. The function N(T(t)) relates temperature to time and finds the time when the bacterial count reaches 1928.

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