Filed the mathematical statements show the statements are true regarding the three options.
[tex]$\begin{array}{l}
\text { and } C=2+1 \\
\text { and } C=8+2 \\
\text { and } C=(8+1)-2
\end{array}$[/tex]
The numerical values of the area and circumference are equal when [tex]$r=2$[/tex].
The numerical value of the area is less than the numerical value of the circumference when [tex]$r\ \textless \ 2$[/tex]
The numerical value of the area is greater than the numerical value of the circumference when [tex]$r\ \textless \ 2$[/tex]
The numerical value of the area is less than the numerical value of the circumference when [tex]$r\ \textgreater \ 2$[/tex].
The numerical value of the area is greater than the numerical value of the circumference when [tex]$r\ \textgreater \ 2$[/tex]
Answer (1)
The area of a circle is A = π r 2 and the circumference is C = 2 π r .
When r = 2 , A = C , so the area and circumference are equal.
When r < 2 , A < C , meaning the area is less than the circumference.
When 2"> r > 2 , C"> A > C , meaning the area is greater than the circumference.
The true statements are: area and circumference are equal when r = 2 , area is less than circumference when r < 2 , and area is greater than circumference when 2"> r > 2 .
C}"> True statements: r = 2 ⇒ A = C , r < 2 ⇒ A 2 ⇒ A > C
Explanation
Problem Analysis and Setup Let's analyze the relationships between the area and circumference of a circle based on its radius. The area of a circle is given by the formula A = π r 2 , and the circumference is given by C = 2 π r , where r is the radius of the circle. We will evaluate the given statements to determine their truthfulness.
Checking the first statement First, let's check the statement: 'The numerical values of the area and circumference are equal when r = 2 '. When r = 2 , the area is A = π ( 2 ) 2 = 4 π , and the circumference is C = 2 π ( 2 ) = 4 π . Since A = C = 4 π when r = 2 , this statement is true.
Analyzing when Area < Circumference Now, let's analyze the relationship between the area and circumference for different values of r . We want to compare A = π r 2 and C = 2 π r .
To determine when the area is less than the circumference ( A < C ), we set up the inequality: π r 2 < 2 π r Divide both sides by π r (assuming 0"> r > 0 ): r < 2 So, the area is less than the circumference when r < 2 . The statement 'The numerical value of the area is less than the numerical value of the circumference when r < 2 ' is true.
Analyzing when Area > Circumference Next, let's consider when the area is greater than the circumference ( C"> A > C ): 2\pi r"> π r 2 > 2 π r Divide both sides by π r (assuming 0"> r > 0 ): 2"> r > 2 So, the area is greater than the circumference when 2"> r > 2 . The statement 'The numerical value of the area is greater than the numerical value of the circumference when 2"> r > 2 ' is true.
Evaluating remaining statements Now we can evaluate the remaining statements:
'The numerical value of the area is greater than the numerical value of the circumference when r < 2 '. This statement is false because we found that A < C when r < 2 .
'The numerical value of the area is less than the numerical value of the circumference when 2"> r > 2 '. This statement is false because we found that C"> A > C when 2"> r > 2 .
Conclusion In summary, the true statements are:
The numerical values of the area and circumference are equal when r = 2 .
The numerical value of the area is less than the numerical value of the circumference when r < 2 .
The numerical value of the area is greater than the numerical value of the circumference when 2"> r > 2 .
Examples
Understanding the relationship between a circle's area and circumference is useful in many real-world scenarios. For example, when designing a circular garden, you might want to ensure that the area covered by plants is larger than the perimeter of the surrounding fence if the radius is greater than 2 units (e.g., meters). Conversely, if you're working with smaller circles (radius less than 2), the length of the border (circumference) will be numerically greater than the planting area. This helps in optimizing material usage and space planning.
