Math Problem Statement

What is A(r(t)), which represents the circle's area after t seconds?

Solution

We are tasked with determining A(r(t))A(r(t)), the area of the circle after tt seconds, using the given formulas:

  1. The radius as a function of tt is r(t)=3+2tr(t) = 3 + 2t.
  2. The area of a circle with radius rr is A(r)=πr2A(r) = \pi r^2.

Step 1: Substitute r(t)r(t) into A(r)A(r)

Using r(t)=3+2tr(t) = 3 + 2t, substitute into the area formula: A(r(t))=π(r(t))2=π(3+2t)2A(r(t)) = \pi \left(r(t)\right)^2 = \pi \left(3 + 2t\right)^2

Step 2: Expand (3+2t)2\left(3 + 2t\right)^2

(3+2t)2=32+2(3)(2t)+(2t)2=9+12t+4t2(3 + 2t)^2 = 3^2 + 2(3)(2t) + (2t)^2 = 9 + 12t + 4t^2

Step 3: Multiply by π\pi

A(r(t))=π(9+12t+4t2)=9π+12πt+4πt2A(r(t)) = \pi (9 + 12t + 4t^2) = 9\pi + 12\pi t + 4\pi t^2

Step 4: Match with the given choices

The correct answer is: D:4πt2+12πt+9π\boxed{D: 4\pi t^2 + 12\pi t + 9\pi}

Let me know if you'd like a breakdown of any specific step or concept.

Questions to Explore:

  1. What are other examples of composite functions like A(r(t))A(r(t)) in real-world applications?
  2. How does changing the function r(t)r(t) affect the area function A(r(t))A(r(t))?
  3. What is the significance of expanding (3+2t)2(3 + 2t)^2 in this problem?
  4. Can you derive the same result using a graphing approach for r(t)r(t)?
  5. How does the concept of function composition apply in calculus?

Tip:

When solving composite function problems, always substitute carefully and simplify systematically!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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