Question

#15 reasoning consider the circles a, b, c, and d. a. without calculating, circle \boxed{} has the greatest circumference. explain. b. without calculating, circle \boxed{} has the least circumference. explain.

Explanation:

Step1: Recall Circumference Formula

The circumference of a circle is given by \( C = \pi d \) (where \( d \) is the diameter) or \( C = 2\pi r \) (where \( r \) is the radius, and \( d = 2r \)). So, a larger diameter (or radius) means a larger circumference.

Step2: Analyze Each Circle's Dimensions

- **Circle A**: Diameter \( d_A = 8 \) ft (since the red line is the diameter).
- **Circle B**: Radius \( r_B = 10 \) m, so diameter \( d_B = 2\times10 = 20 \) m. Convert to feet (1 m ≈ 3.28 ft), \( d_B \approx 20\times3.28 = 65.6 \) ft.
- **Circle C**: Diameter \( d_C = 2.5 \) m. Convert to feet: \( 2.5\times3.28 = 8.2 \) ft.
- **Circle D**: Radius \( r_D = 50 \) in. Convert to feet (1 in = \( \frac{1}{12} \) ft), \( r_D = \frac{50}{12} \approx 4.17 \) ft, so diameter \( d_D = 2\times4.17 \approx 8.34 \) ft. Wait, no—wait, maybe I misread. Wait, Circle B: the label is "10 m" as radius? Wait, no, the blue circle (B) has a radius of 10 m? Wait, no, maybe the units: let's check again. Wait, maybe the problem has mixed units, but we can compare diameters in the same unit or just by relative size. Wait, no, maybe I made a mistake. Wait, Circle A: diameter 8 ft. Circle B: radius 10 m (so diameter 20 m = 65.6 ft). Circle C: diameter 2.5 m = 8.2 ft. Circle D: radius 50 in = 50/12 ≈ 4.17 ft, diameter ≈ 8.34 ft. Wait, no, that can't be. Wait, maybe Circle B's radius is 10 m, so diameter 20 m, which is much larger than 8 ft. Circle D: radius 50 in is about 4.17 ft, diameter ~8.34 ft. Circle C: 2.5 m is 8.2 ft. Circle A: 8 ft. Wait, but maybe the units are all in the same? No, the problem has ft, m, in. But the key is that circumference is proportional to diameter (or radius). So, the largest diameter (or radius) will have the largest circumference. Circle B has a radius of 10 m, which is a very large diameter (20 m) compared to the others. For the least circumference, we need the smallest diameter. Circle A: 8 ft, Circle C: 2.5 m ≈ 8.2 ft, Circle D: ~8.34 ft, Circle B: 20 m. Wait, no, that's not right. Wait, maybe I misread Circle B: maybe the 10 m is diameter? Wait, the blue circle (B) has a radius marked? Wait, the image: Circle A (yellow) has a diameter of 8 ft (red line through center). Circle B (blue) has a radius of 10 m (blue dot is center, red line is radius? Wait, no, the red line in B is from center to edge, so radius 10 m. Circle C (green) has a diameter of 2.5 m (red line through center). Circle D (purple) has a radius of 50 in (red line from center to edge).

Wait, let's re-express all diameters in feet:

- Circle A: \( d_A = 8 \) ft.
- Circle B: \( d_B = 2\times10 \) m = 20 m. 1 m = 3.28084 ft, so \( d_B = 20\times3.28084 \approx 65.6168 \) ft.
- Circle C: \( d_C = 2.5 \) m = \( 2.5\times3.28084 \approx 8.2021 \) ft.
- Circle D: \( d_D = 2\times50 \) in = 100 in. 1 in = 0.0833333 ft, so \( d_D = 100\times0.0833333 \approx 8.3333 \) ft.

Now, compare diameters: \( d_B \approx 65.6 \) ft (largest), \( d_D \approx 8.33 \) ft, \( d_C \approx 8.2 \) ft, \( d_A = 8 \) ft (smallest? Wait, no, 8 ft is smaller than 8.2 and 8.33. Wait, so Circle A has diameter 8 ft, which is smaller than Circle C (8.2 ft) and D (8.33 ft), and B is much larger. Wait, but the question a is "which circle has the greatest circumference". Since circumference is \( \pi d \), the largest diameter is Circle B (65.6 ft), so Circle B has the greatest circumference. For part b, the smallest diameter is Circle A (8 ft), so Circle A has the least circumference? Wait, but my earlier calculation for Circle C was 2.5 m = 8.2 ft, which is larger than 8 ft. Circle D: 100 in = 8.33 ft, also larger than 8 ft. So Circle A has the smallest diameter, hence least circumference.

Wait, but maybe I messed up Circle B's radius. Wait, maybe Circle B's radius is 10 m, so diameter 20 m, which is way larger than the others. So for part a, Circle B has the greatest circumference. For part b, Circle A has the least circumference.

Answer (Part a):

Circle B has the greatest circumference because its diameter (20 m, or ~65.6 ft) is much larger than the diameters of the other circles (8 ft, ~8.2 ft, ~8.33 ft), and circumference is proportional to diameter (\( C = \pi d \)).

Answer (Part b):

Circle A has the least circumference because its diameter (8 ft) is smaller than the diameters of the other circles (Circle B: ~65.6 ft, Circle C: ~8.2 ft, Circle D: ~8.33 ft), and circumference is proportional to diameter (\( C = \pi d \)).

(Note: If the problem expects the circle labels, e.g., "Circle B" for part a and "Circle A" for part b, adjust accordingly. The key is using the circumference formula's proportionality to diameter/radius.)

Answer:

(Part a):
Circle B has the greatest circumference because its diameter (20 m, or ~65.6 ft) is much larger than the diameters of the other circles (8 ft, ~8.2 ft, ~8.33 ft), and circumference is proportional to diameter (\( C = \pi d \)).