Math Problem Statement
Solution
We are tasked with determining , the area of the circle after seconds, using the given formulas:
- The radius as a function of is .
- The area of a circle with radius is .
Step 1: Substitute into
Using , substitute into the area formula:
Step 2: Expand
Step 3: Multiply by
Step 4: Match with the given choices
The correct answer is:
Let me know if you'd like a breakdown of any specific step or concept.
Questions to Explore:
- What are other examples of composite functions like in real-world applications?
- How does changing the function affect the area function ?
- What is the significance of expanding in this problem?
- Can you derive the same result using a graphing approach for ?
- How does the concept of function composition apply in calculus?
Tip:
When solving composite function problems, always substitute carefully and simplify systematically!
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Math Problem Analysis
Mathematical Concepts
Function Composition
Geometry
Quadratic Expressions
Formulas
r(t) = 3 + 2t
A(r) = πr²
Theorems
-
Suitable Grade Level
Grades 9-12
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