Question

1. you are using a math software program to design a pattern for an olympic flag. in addition to the dimensions shown in the diagram, the distance between the outer edges of any two adjacent rings in the same row is 3 inches.

a. use the given dimensions to write equations representing the outer circles of the five rings. use inches as units in a coordinate plane with the lower left corner of the flag on the origin.

b. each ring is 3 inches thick. explain how you can adjust the equations of the outer circles to write equations representing the inner circles.

Explanation:

Step1: Find radius of outer circles

The diameter of each outer ring is 15 in, so radius $r = \frac{15}{2} = 7.5$ in.

Step2: Locate center of bottom-left circle

Center coordinates: $x=15$, $y=15$ (from origin, half the diameter).
Equation: $(x-15)^2 + (y-15)^2 = 7.5^2$

Step3: Locate center of bottom-middle circle

Horizontal distance between adjacent outer edges is 3 in, so center $x=15+7.5+3+7.5=33$, $y=15$.
Equation: $(x-33)^2 + (y-15)^2 = 7.5^2$

Step4: Locate center of top-left circle

Vertical distance from bottom row: $18-7.5-7.5=3$ in, so center $x=26$, $y=15+7.5+3+7.5=33$.
Equation: $(x-26)^2 + (y-33)^2 = 7.5^2$

Step5: Locate center of top-middle circle

Center $x=26+7.5+3+7.5=44$, $y=33$.
Equation: $(x-44)^2 + (y-33)^2 = 7.5^2$

Step6: Locate center of top-right circle

Center $x=44+7.5+3+7.5=62$, $y=33$.
Equation: $(x-62)^2 + (y-33)^2 = 7.5^2$

Step7: Adjust for inner circles (part b)

Inner radius $r_{inner}=7.5-3=4.5$ in. Use the same center coordinates as outer circles, substitute radius to get inner circle equations.

Answer:

Part a: Outer Circle Equations

1. Bottom-left: $(x-15)^2 + (y-15)^2 = 7.5^2$
2. Bottom-middle: $(x-33)^2 + (y-15)^2 = 7.5^2$
3. Top-left: $(x-26)^2 + (y-33)^2 = 7.5^2$
4. Top-middle: $(x-44)^2 + (y-33)^2 = 7.5^2$
5. Top-right: $(x-62)^2 + (y-33)^2 = 7.5^2$

Part b: Inner Circle Adjustment

Each ring is 3 inches thick, so the radius of each inner circle is the outer radius minus 3: $7.5-3=4.5$ inches. Use the exact same center coordinates as the corresponding outer circle, and replace the radius term in the circle equation with $4.5^2$. The inner circle equations are:

1. Bottom-left: $(x-15)^2 + (y-15)^2 = 4.5^2$
2. Bottom-middle: $(x-33)^2 + (y-15)^2 = 4.5^2$
3. Top-left: $(x-26)^2 + (y-33)^2 = 4.5^2$
4. Top-middle: $(x-44)^2 + (y-33)^2 = 4.5^2$
5. Top-right: $(x-62)^2 + (y-33)^2 = 4.5^2$