(1 point) The number of bacteria in a refrigerated food product is given by N(T)=24T2−76T+47, 3<T<33 where T is the temperature of the food.
When the food is removed from the refrigerator, the temperature is given by T(t)=7t+1.7 , where t is the time in hours.
Find the composite function N(T(t)):
N(T(t))=
Find the time when the bacteria count reaches 2163
Time Needed =

### Step 1: Find the composite function \( N(T(t)) \)

We are given the following functions:
- \( N(T) = 24T^2 - 76T + 47 \)
- \( T(t) = 7t + 1.7 \)

To find the composite function \( N(T(t)) \), we substitute \( T(t) \) into \( N(T) \):

\[
N(T(t)) = 24(7t + 1.7)^2 - 76(7t + 1.7) + 47
\]

Now, expand and simplify.

#### Squaring \( (7t + 1.7) \):
\[
(7t + 1.7)^2 = 49t^2 + 2(7t)(1.7) + (1.7)^2 = 49t^2 + 23.8t + 2.89
\]

Now, substitute this into the equation for \( N(T(t)) \):

\[
N(T(t)) = 24(49t^2 + 23.8t + 2.89) - 76(7t + 1.7) + 47
\]

Expand each term:

\[
N(T(t)) = 24(49t^2) + 24(23.8t) + 24(2.89) - 76(7t) - 76(1.7) + 47
\]
\[
N(T(t)) = 1176t^2 + 571.2t + 69.36 - 532t - 129.2 + 47
\]

Now, combine like terms:

\[
N(T(t)) = 1176t^2 + (571.2t - 532t) + (69.36 - 129.2 + 47)
\]
\[
N(T(t)) = 1176t^2 + 39.2t - 12.84
\]

So the composite function is:

\[
N(T(t)) = 1176t^2 + 39.2t - 12.84
\]

---

### Step 2: Find the time when the bacteria count reaches 2163

We are asked to find the time \( t \) when \( N(T(t)) = 2163 \). So, set the composite function equal to 2163:

\[
1176t^2 + 39.2t - 12.84 = 2163
\]

Rearrange the equation:

\[
1176t^2 + 39.2t - 2175.84 = 0
\]

Now, solve this quadratic equation using the quadratic formula:

\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here:
- \( a = 1176 \)
- \( b = 39.2 \)
- \( c = -2175.84 \)

Let me calculate the discriminant and solve for \( t \).The solutions for the time \( t \) are approximately:

\[
t_1 \approx 1.34 \text{ hours}, \quad t_2 \approx -1.38 \text{ hours}
\]

Since time cannot be negative, the valid solution is:

\[
t \approx 1.34 \text{ hours}
\]

Thus, the bacteria count reaches 2163 approximately **1.34 hours** after the food is removed from the refrigerator.

Would you like further details or have any questions?

Here are 5 related questions you might find interesting:
1. How would the bacteria count change if the temperature model \( T(t) \) were altered?
2. What if the bacteria threshold was 3000 instead of 2163? How would the time change?
3. Can you generalize a formula for finding when the bacteria count reaches any given value?
4. How sensitive is the bacteria count to changes in the temperature over time?
5. What is the significance of the parabola shape in the quadratic bacteria model?

**Tip:** Always check the validity of solutions, especially when dealing with real-world quantities like time.

The number of bacteria in a refrigerated food product is given by N(T)=24T^2−76T+47, where T is the temperature. When the food is removed from the refrigerator, T(t)=7t+1.7. Find the composite function N(T(t)) and the time when the bacteria count reaches 2163.

Explore how to find the composite function N(T(t)) for a bacteria count model and solve for the time when the bacteria count reaches 2163 using quadratic equations. Detailed step-by-step solution ideal for high school students in Grades 10-12.

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