The area of a circle is a function equal to the product of pi ([tex]$\pi$[/tex]) and the square of the radius ([tex]$r$[/tex]). Which of the following shows this function?
A. [tex]$f(r)=\pi r^2$[/tex]
B. [tex]$f(r)=2 \pi r$[/tex]
C. [tex]$f(r)=2 \pi+2 r^2$[/tex]
D. [tex]$f(r)=\pi^2-r^2$[/tex]
Answer (1)
The problem defines the area of a circle as a function of its radius.
The area A is expressed as A = π r 2 .
The function representing the area is f ( r ) = π r 2 .
The correct option is f ( r ) = π r 2 .
Explanation
Problem Analysis The problem states that the area of a circle is a function of its radius, specifically the product of π and the square of the radius r . We need to identify the correct function from the given options.
Area as a Function of Radius The area A of a circle is given by the formula: A = π r 2 We can represent this area as a function of the radius r :
f ( r ) = π r 2
Comparing with Options Now, let's compare this function with the given options:
Option A: f ( r ) = π r 2 - This matches our derived function. Option B: f ( r ) = 2 π r - This represents the circumference of a circle, not the area. Option C: f ( r ) = 2 π + 2 r 2 - This is not a standard formula related to circles. Option D: f ( r ) = π 2 − r 2 - This is also not a standard formula related to circles.
Final Answer Therefore, the correct function representing the area of a circle is: f ( r ) = π r 2 This corresponds to option A.
Examples
Understanding the area of a circle is crucial in many real-world applications. For instance, when designing a circular garden, you need to calculate the area to determine how much soil or fertilizer is required. Similarly, architects use this formula to calculate the floor area of circular rooms or the surface area of dome-shaped structures. In essence, knowing the area of a circle helps in efficient planning and resource allocation in various fields.
