Question

1. select all the pairs of quantities that are proportional to each other. for the quantities that are proportional, write an equation that relates them.

a. the radius and diameter measurements of a circle
b. the radius and circumference measurements of a circle
c. the radius and area measurements of a circle
d. the diameter and circumference measurements of a circle
e. the diameter and area measurements of a circle

Explanation:

Step1: Recall Proportional Relationships

Proportional relationships have the form \( y = kx \) (direct proportion), where \( k \) is a constant. For circles:

• Diameter \( d = 2r \) (so \( d = 2r \), constant \( k = 2 \), proportional).
• Circumference \( C = 2\pi r \) or \( C=\pi d \) (proportional, \( k = 2\pi \) or \( k=\pi \)).
• Area \( A=\pi r^{2} \) (not proportional to \( r \) or \( d \), since it's a square relationship).

Step2: Analyze Each Option

• Option A: Radius (\( r \)) and diameter (\( d \)): \( d = 2r \), so \( \frac{d}{r}=2 \) (constant). Proportional. Equation: \( d = 2r \).
• Option B: Radius (\( r \)) and circumference (\( C \)): \( C = 2\pi r \), so \( \frac{C}{r}=2\pi \) (constant). Proportional. Equation: \( C = 2\pi r \).
• Option C: Radius (\( r \)) and area (\( A \)): \( A=\pi r^{2} \), \( \frac{A}{r}=\pi r \) (not constant, depends on \( r \)). Not proportional.
• Option D: Diameter (\( d \)) and circumference (\( C \)): \( C=\pi d \), so \( \frac{C}{d}=\pi \) (constant). Proportional. Equation: \( C = \pi d \).
• Option E: Diameter (\( d \)) and area (\( A \)): \( A=\pi (\frac{d}{2})^{2}=\frac{\pi}{4}d^{2} \), \( \frac{A}{d}=\frac{\pi}{4}d \) (not constant). Not proportional.

Answer:

A. The radius and diameter measurements of a circle, \( d = 2r \)
B. The radius and circumference measurements of a circle, \( C = 2\pi r \)
D. The diameter and circumference measurements of a circle, \( C = \pi d \)