Math Problem Statement
Solution
Let's break down the question step by step:
Given:
- The concentration of the drug in the patient's body is modeled by the quadratic function: where is the concentration in mg/L, and is the time in hours after injection.
Part A: Domain and Range
We need to find the domain and range of the function based on the context of the problem.
-
Domain:
- The domain represents the possible values of , which is the time after the medication is injected.
- Time cannot be negative, so .
- The function is a quadratic opening downwards (since the coefficient of is negative), meaning there is a maximum point. After a certain point, the concentration decreases back to 0.
- We need to find when to determine when the concentration reaches 0 again.
- Solve : Solutions: or .
Therefore, the domain is .
-
Range:
- The range depends on the maximum value of . Since it's a downward-opening parabola, we need to find the vertex to determine the maximum value.
- The vertex occurs at , where and :
- Now, find :
- Therefore, the maximum concentration is 5 mg/L, and since starts at 0 and reaches 0 again at , the range is .
Summary for Part A:
- Domain: hours
- Range: mg/L
Part B: Graphing and Maximum Concentration
The graph of is a parabola that opens downwards. Based on Part A, the vertex is at , where the maximum concentration is 5 mg/L. The graph starts at with , peaks at with , and returns to 0 at .
Maximum concentration: 5 mg/L at hour.
Would you like a more detailed explanation of any step or the graph plotted for better understanding?
Here are 5 related questions to deepen your understanding:
- How do you calculate the vertex of a parabola algebraically?
- Why is the domain limited to hours?
- How can you determine the maximum point of any quadratic function?
- What real-world scenarios might exhibit a quadratic concentration pattern like this?
- How does the sign of the quadratic term affect the shape of the parabola?
Tip: For any quadratic function , the maximum or minimum occurs at the vertex, and you can find it using .
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Functions
Graphing Parabolas
Formulas
C(t) = -5t^2 + 10t
Vertex formula: t = -b / (2a)
Quadratic roots
Solving C(t) = 0
Theorems
Quadratic function and its maximum point theorem
Domain and range for real-world contexts
Suitable Grade Level
Grades 9-12
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