Question

the circumference of circle t is 4 times the circumference of circle s. choose all of the pairs of circles that show possible radii for circle s and circle t. (note: images are not drawn to scale)

Explanation:

Step1: Recall Circumference Formula

The circumference of a circle is given by \( C = 2\pi r \), where \( r \) is the radius. If the circumference of circle \( T \) (\( C_T \)) is 4 times the circumference of circle \( S \) (\( C_S \)), then \( C_T = 4C_S \). Substituting the formula for circumference, we get \( 2\pi r_T = 4\times(2\pi r_S) \).

Step2: Simplify the Equation

Simplify \( 2\pi r_T = 8\pi r_S \) by dividing both sides by \( 2\pi \). This gives \( r_T = 4r_S \), meaning the radius of circle \( T \) should be 4 times the radius of circle \( S \).

Step3: Check Each Pair

• First pair: \( r_S = 2 \), \( r_T = 4 \). \( 4\div2 = 2

eq4 \). Not valid.

• Second pair: \( r_S = 2 \), \( r_T = 8 \). \( 8\div2 = 4 \). Valid.
• Third pair: \( r_S = 2 \), \( r_T = 32 \). \( 32\div2 = 16

eq4 \). Not valid.

• Fourth pair: \( r_S = 4 \), \( r_T = 16 \). \( 16\div4 = 4 \). Valid.
• Fifth pair: \( r_S = 8 \), \( r_T = 16 \). \( 16\div8 = 2

eq4 \). Not valid.

Answer:

The pairs with \( r_S = 2, r_T = 8 \) and \( r_S = 4, r_T = 16 \) (the second and fourth boxes) are the correct ones. So we check the boxes for the pairs where \( S \) has radius 2 and \( T \) has radius 8, and \( S \) has radius 4 and \( T \) has radius 16.