Question

finding circumferences find the circumferences of both circles. 21. circle diagram with inner radius r and outer radius 5 cm 22. circle diagram with inner diameter 9 ft and outer radius 2.5 ft 23. partial circle diagram with 22...

Explanation:

Problem 21

Step1: Identify radii of circles

Inner circle radius \( r_1 = 5 \, \text{cm} \), outer circle radius \( r_2 = 5 + 5 = 10 \, \text{cm} \) (since the distance from inner center to outer edge is 5 cm, so outer radius is inner radius + 5 cm).

Step2: Circumference formula \( C = 2\pi r \)

For inner circle: \( C_1 = 2\pi r_1 = 2\pi(5) = 10\pi \approx 31.4 \, \text{cm} \)
For outer circle: \( C_2 = 2\pi r_2 = 2\pi(10) = 20\pi \approx 62.8 \, \text{cm} \)

Problem 22

Step1: Identify diameters/radii

Inner circle diameter \( d_1 = 9 \, \text{ft} \), so radius \( r_1 = \frac{9}{2} = 4.5 \, \text{ft} \). Outer circle radius \( r_2 = 4.5 + 2.5 = 7 \, \text{ft} \) (since the width of the ring is 2.5 ft, so outer radius = inner radius + 2.5 ft).

Step2: Circumference formula \( C = 2\pi r \) or \( C = \pi d \)

For inner circle (using \( C = \pi d \)): \( C_1 = \pi d_1 = 9\pi \approx 28.26 \, \text{ft} \)
For outer circle: \( C_2 = 2\pi r_2 = 2\pi(7) = 14\pi \approx 43.96 \, \text{ft} \)

Problem 23 (Assuming the given is diameter 22, let's solve)

Step1: Identify diameter

Diameter \( d = 22 \) (assuming unit, say, units). Radius \( r = \frac{22}{2} = 11 \). If there's an outer circle, but since only 22 is given (maybe diameter of inner, and we need to find both? Wait, the diagram shows a ring, so similar to 21 and 22. Wait, maybe the outer radius is inner radius + something? Wait, the given is 22 (maybe diameter of inner, and the ring width? Wait, the original problem for 23: let's assume the inner diameter is 22, and the ring width is, say, let's check the pattern. Wait, maybe the 22 is diameter of inner, and outer radius is inner radius + some? Wait, maybe the problem is similar: inner diameter 22, so inner radius 11, and outer radius 11 + x? Wait, maybe the diagram has a ring with width, but since it's cut off, maybe we assume it's a ring with inner diameter 22, and outer radius = inner radius + width. But since it's not clear, maybe the user missed. But assuming it's a circle (or ring), let's proceed with inner circle:

Step1: Diameter \( d = 22 \), radius \( r = 11 \)

Step2: Circumference \( C = \pi d = 22\pi \approx 69.08 \) or \( C = 2\pi r = 22\pi \)

Answer:

s:

Problem 21:

Inner circle circumference: \( \boldsymbol{10\pi \, \text{cm} \approx 31.4 \, \text{cm}} \)
Outer circle circumference: \( \boldsymbol{20\pi \, \text{cm} \approx 62.8 \, \text{cm}} \)

Problem 22:

Inner circle circumference: \( \boldsymbol{9\pi \, \text{ft} \approx 28.26 \, \text{ft}} \)
Outer circle circumference: \( \boldsymbol{14\pi \, \text{ft} \approx 43.96 \, \text{ft}} \)

Problem 23 (assuming inner diameter 22):

Inner circle circumference: \( \boldsymbol{22\pi \approx 69.08} \) (units, e.g., if diameter is 22 units)
(If outer circle, need more info, but based on pattern, maybe outer radius = 11 + width, but since not given, we can only do inner with diameter 22)