Horatio drew a circle with a radius of [tex]$6x \text{ cm}$[/tex]. Kelli drew a circle with a radius of [tex]$18y \text{ cm}$[/tex]. What is the scale factor between these two circles?
A. [tex]$\frac{x}{3 y}$[/tex]
B. [tex]$\frac{2 x}{3 y}$[/tex]
C. [tex]$\frac{1}{3}$[/tex]
D. [tex]$\frac{2}{3}$[/tex]
Answer (1)
Set up the ratio of the radii of the two circles: 18 y 6 x .
Simplify the fraction by dividing both the numerator and the denominator by 6: 18 y ÷ 6 6 x ÷ 6 = 3 y x .
The scale factor between the two circles is 3 y x .
Explanation
Problem Analysis Let's analyze the problem. We are given the radii of two circles, one drawn by Horatio and the other by Kelli. We need to find the scale factor between these two circles. The scale factor is the ratio of the radius of Horatio's circle to the radius of Kelli's circle.
Given Information Horatio's circle has a radius of 6 x cm. Kelli's circle has a radius of 18 y cm.
Setting up the Ratio The scale factor is the ratio of Horatio's radius to Kelli's radius. So, we set up the ratio as follows: Kelli’s radius Horatio’s radius = 18 y 6 x
Simplifying the Fraction Now, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6: 18 y 6 x = 18 y ÷ 6 6 x ÷ 6 = 3 y x
Final Answer Therefore, the scale factor between the two circles is 3 y x .
Examples
Scale factors are used in many real-world applications, such as mapmaking. A map is a scaled-down version of a real geographical area. If a map has a scale factor of 1:100,000, it means that 1 cm on the map represents 100,000 cm (or 1 km) in the real world. Understanding scale factors helps us to accurately represent and interpret sizes and distances in various contexts, from architectural blueprints to model building.
