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Let's solve the questions provided in the image step by step.

### Given:
1. The total cost of manufacturing $ q $ units is:
   \[
   C(q) = q^2 + q + 900
   \]
2. The number of units manufactured after $ t $ hours is modeled by:
   \[
   q(t) = 25t
   \]

#### a. **Express the total manufacturing cost as a function of $ t $.**

To express the cost $ C $ as a function of $ t $, we substitute $ q(t) = 25t $ into the cost function $ C(q) $:

\[
C(t) = C(q(t)) = (25t)^2 + (25t) + 900
\]

Simplifying:

\[
C(t) = 625t^2 + 25t + 900
\]

So, the total manufacturing cost as a function of $ t $ is:
\[
C(t) = 625t^2 + 25t + 900
\]

#### b. **How much will have been spent on production by the end of the third hour?**

We substitute $ t = 3 $ into the cost function $ C(t) $:

\[
C(3) = 625(3)^2 + 25(3) + 900
\]

Calculating step by step:

\[
C(3) = 625 \times 9 + 75 + 900
\]
\[
C(3) = 5625 + 75 + 900
\]
\[
C(3) = 6600
\]

So, the total cost spent by the end of the third hour is **$6,600**.

#### c. **When will the total manufacturing cost reach $11,000?**

We set $ C(t) = 11000 $ and solve for $ t $:

\[
625t^2 + 25t + 900 = 11000
\]

First, subtract 11000 from both sides:

\[
625t^2 + 25t + 900 - 11000 = 0
\]
\[
625t^2 + 25t - 10100 = 0
\]

Now, divide through by 25 to simplify:

\[
25t^2 + t - 404 = 0
\]

We can solve this quadratic equation using the quadratic formula:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where $ a = 25 $, $ b = 1 $, and $ c = -404 $.

Let's calculate this value.The solutions to the quadratic equation are:

\[
t = 4.0 \quad \text{or} \quad t = -4.04
\]

Since negative time doesn't make sense in this context, we conclude that the total manufacturing cost will reach **$11,000** after **4 hours**.

---

Would you like any further details or clarification?

Here are 5 related questions for further exploration:
1. How would changes in the production rate $ q(t) $ affect the total cost function?
2. What would the cost be after 5 hours of production?
3. Can you derive a general formula for the number of hours required to reach a given total cost?
4. How much of the total cost is fixed, and how much depends on the units produced?
5. How would the cost equation change if the fixed cost of 900 dollars increased by 10%?

**Tip**: When dealing with quadratic equations, always check if dividing the equation by a common factor simplifies calculations.

At a certain factory, the total cost of manufacturing q units during the daily production run is C(q) = q^2 + q + 900 dollars. On a typical workday, the number of units manufactured during the first t hours of a production run can be modeled by the function q(t) = 25t. a. Express the total manufacturing cost as a function of t. b. How much will have been spent on production by the end of the third hour? c. When will the total manufacturing cost reach $11,000?

Learn how to express the total manufacturing cost as a function of time using quadratic equations. This example explores calculating costs after a given time and determining when a cost reaches a specific value. Suitable for high school students in Grades 10-12.

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