Since all circles are similar, a proportion can be set up using the circumference and diameter of each circle. Substitute the values [tex]$d_1=1, C_1=\pi$[/tex], and [tex]$d_2=2 r$[/tex] into the proportion.
[tex]$\frac{C_1}{d_1}=\frac{C_2}{d_2}$[/tex]
Which shows how to correctly solve for [tex]$C_2$[/tex], the circumference of any circle with radius [tex]$r$[/tex]?
A. Because [tex]$\frac{\pi}{1}=\frac{C_2}{2 r}, C_2=2 r \pi$[/tex]
B. Because [tex]$\frac{1}{\pi}=\frac{C_2}{2 r}, C_2=\frac{2 r}{\pi}$[/tex]
C. Because [tex]$\frac{\pi}{2 r}=\frac{C_2}{1}, C_2=\frac{\pi}{2 r}$[/tex]
D. Because [tex]$\frac{\pi}{1}=\frac{C_2}{4 r}, C_2=4 r \pi$[/tex]
Answer (2)
To find the circumference C 2 of a circle, we substitute C 1 = π , d 1 = 1 , and d 2 = 2 r into the proportion and solve. This yields the result C 2 = 2 π r . The correct answer is option A.
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Substitute the given values d 1 = 1 , C 1 = π , and d 2 = 2 r into the proportion d 1 C 1 = d 2 C 2 .
Obtain the equation 1 π = 2 r C 2 .
Multiply both sides by 2 r to isolate C 2 .
Find the circumference: C 2 = 2 π r .
Explanation
Set up the proportion We are given the proportion d 1 C 1 = d 2 C 2 , and we are given the values d 1 = 1 , C 1 = π , and d 2 = 2 r . We want to solve for C 2 .
Substitute the given values Substitute the given values into the proportion: 1 π = 2 r C 2
Solve for C_2 To solve for C 2 , multiply both sides of the equation by 2 r :
C 2 = 2 r × 1 π C 2 = 2 π r
State the final answer Therefore, the correct expression for C 2 is C 2 = 2 π r .
Examples
Understanding the relationship between a circle's radius and circumference is useful in many real-world scenarios. For example, if you're designing a circular garden and need to determine how much fencing to buy, you can use the formula C = 2 π r to calculate the circumference (the length of the fence) based on the radius of the garden. Similarly, if you're a mechanical engineer designing a circular gear, knowing the relationship between radius and circumference helps you calculate the gear's size and how it interacts with other gears. This principle applies to anything circular, from wheels to pizzas!
