Question

A jeweler resizes a ring so that its inner circumference is 4 centimeters. (a) What is the radius r (in cm ) of the ring? r=cm (b) The inner circumference of the ring varies between 3.5 centimeters and 4.5 centimeters. How does the radius vary?

Answer 1

Using the formula for the circumference of a circle (C = 2πr), the radius r of a ring with a 4 cm circumference is approximately 0.6366 cm. As the circumference varies from 3.5 cm to 4.5 cm, the radius changes from approximately 0.5572 cm to 0.7162 cm.

Step-by-step explanation:

The student's question involves finding the radius of a ring given its circumference and understanding how the radius varies with changes in the circumference. Let's solve each part step by step.

Part A: Finding the radius r (in cm) of the ring:

The formula for the circumference (C) of a circle is:

C = 2πr

Where π (pi) is approximately 3.14159 and r is the radius. Given that the circumference of the ring is 4 centimeters, we can find the radius by rearranging the formula:

r = C / (2π)

Substituting the given values:

r = 4 cm / (2 x 3.14159)

Doing the calculation gives us:

r ≈ 0.6366 cm

Part B: Variation of the radius with the circumference

Similarly, we can calculate the radius when the circumference varies between 3.5 cm and 4.5 cm.

For C = 3.5 cm:

r = 3.5 cm / (2 x 3.14159) ≈ 0.5572 cm

For C = 4.5 cm:

r = 4.5 cm / (2 x 3.14159) ≈ 0.7162 cm

Therefore, the radius varies between approximately 0.5572 cm and 0.7162 cm as the inner circumference changes from 3.5 cm to 4.5 cm.