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Answer part B and generate a graph that corresponds with the answer

For Part B:

The greatest concentration of the medication occurs when \( t = 1 \) hour, and at that time, the concentration \( C(1) \) is \( 5 \, \text{mg/L} \).

Here is the graph of the function \( C(t) = -5t^2 + 10t \). It shows the concentration increasing to a maximum of \( 5 \, \text{mg/L} \) at \( t = 1 \, \text{hour} \), and then decreasing as time continues. The graph highlights this peak with a red marker at the maximum concentration point.

If you have further questions or need more details, feel free to ask!

Here are some related questions you might want to explore:
1. What is the significance of the negative coefficient for \( t^2 \) in the concentration formula?
2. How would the graph change if the coefficients of the function were different?
3. Could the function \( C(t) \) ever take on negative values? Why or why not?
4. How would you interpret the rate of change of concentration in terms of the function's derivative?
5. What practical implications might the shape of this graph have for medication dosage scheduling?

**Tip:** When determining maximum or minimum values of a quadratic function, completing the square or using the vertex formula \( t = -\frac{b}{2a} \) is a helpful approach.

Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body, and the time when that occurs.

Learn how to graph and analyze the quadratic function C(t) = -5t^2 + 10t, which models the concentration of a medication in a patient's body over time. Determine the time and value of maximum concentration. Suitable for Grades 10-12.

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