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The problem shown in the image asks to find the time when the bacteria count reaches 19801, given the function for bacteria count \( N(T) = 30T^2 - 97T + 56 \), where \( T \) is the temperature of the food. The temperature as a function of time (in hours) is given by \( T(t) = 9t + 1.6 \).

To find the time when the bacteria count reaches 19801, we first need to substitute the expression for \( T(t) \) into \( N(T) \) to form a composite function \( N(T(t)) \), and then solve for \( t \).

### Given:
1. \( N(T) = 30T^2 - 97T + 56 \)
2. \( T(t) = 9t + 1.6 \)

The composite function \( N(T(t)) \) is:
\[
N(T(t)) = 30(9t + 1.6)^2 - 97(9t + 1.6) + 56
\]

You are asked to find the time \( t \) when \( N(T(t)) = 19801 \).

### Steps:
1. Substitute \( T(t) = 9t + 1.6 \) into \( N(T) \).
2. Expand and simplify the expression.
3. Set \( N(T(t)) = 19801 \).
4. Solve the resulting quadratic equation for \( t \).

I'll go ahead and solve this equation for you.The solutions to the equation are approximately \( t = -2.85 \) and \( t = 2.86 \). Since time cannot be negative in this context, the valid solution is:

\[
t \approx 2.86 \, \text{hours}
\]

Thus, the bacteria count will reach 19801 after about **2.86 hours**.

Would you like more details on any of these steps?

Here are 5 related questions you might find interesting:
1. How is the composite function \( N(T(t)) \) derived?
2. Why are negative values of time not valid in this context?
3. Can this method be applied to other temperature-dependent growth models?
4. How does changing the initial temperature affect the time needed?
5. What is the significance of quadratic growth in bacterial populations?

**Tip:** When solving real-world equations, always check the feasibility of your solutions, especially when dealing with time or quantities that must be positive.

Find the time when the bacteria count reaches 19801, given the function N(T) = 30T^2 - 97T + 56, where T is the temperature, and T(t) = 9t + 1.6 represents temperature as a function of time.

Learn how to solve a composite function problem involving bacterial growth with a quadratic equation. This tutorial walks through solving N(T) = 30T^2 - 97T + 56 and using the temperature function T(t) = 9t + 1.6 to find when the bacteria count reaches 19801. Suitable for students in Grades 9-12.

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