The diameter of one circle is represented by [tex]$12 x$[/tex]. The diameter of another circle is represented by [tex]$6 x^2 y$[/tex]. What is the ratio of the radii of the two circles?
Answer (1)
The problem involves finding the ratio of the radii of two circles given their diameters.
First, find the radius of each circle by dividing its diameter by 2: r 1 = 6 x and r 2 = 3 x 2 y .
Then, find the ratio of the radii: r 2 r 1 = 3 x 2 y 6 x .
Finally, simplify the ratio: x y 2 .
Express the ratio in the form a : b : 2 : x y .
Explanation
Problem Analysis Let's analyze the problem. We are given the diameters of two circles and we need to find the ratio of their radii.
Radius of the first circle The diameter of the first circle is 12 x . Therefore, the radius of the first circle, r 1 , is half of the diameter: r 1 = 2 12 x = 6 x
Radius of the second circle The diameter of the second circle is 6 x 2 y . Therefore, the radius of the second circle, r 2 , is half of the diameter: r 2 = 2 6 x 2 y = 3 x 2 y
Ratio of the radii Now, we need to find the ratio of the radii, which is r 2 r 1 :
r 2 r 1 = 3 x 2 y 6 x
Simplify the ratio Simplify the ratio by dividing both the numerator and the denominator by their common factors. Both 6 x and 3 x 2 y are divisible by 3 x :
3 x 2 y 6 x = 3 x 2 y ÷ 3 x 6 x ÷ 3 x = x y 2
Final ratio Express the ratio in the form a : b :
2 : x y
Examples
In architecture, when designing circular structures like domes or arches, understanding the ratio of radii is crucial for maintaining structural integrity and aesthetic balance. For example, if you're designing two concentric circular arches, knowing the ratio of their radii helps ensure that the load is distributed evenly, preventing stress concentrations and potential collapses. This principle applies to various engineering and design scenarios where circular elements interact.
