- Establish the relationship \(\frac{C}{d} \approx \pi \approx 3.14\).
- Calculate the ratios for each pair: \(\text{A: } 4.4\), \(\text{B: } \approx 3.158\), \(\text{C: } \approx 2.424\).
- Compare the ratios to 3.14.
- Select the pair with the ratio closest to 3.14, which is \(\boxed{B}\).
Explanation
- Analysis of the geometric relationship between diameter and circumference.
To determine which pairs \((d, C)\) represent a reasonable approximation for the diameter \(d\) and circumference \(C\) of a circle, we use the fundamental geometric relationship \(C = \pi d\). This implies that the ratio \(\frac{C}{d}\) must be approximately equal to the value of \(\pi \approx 3.14159\). We will calculate this ratio for each provided pair and check how close it is to \(\pi\).
- Calculation of the ratios for each pair.
For each pair, we calculate the ratio \(\frac{C}{d}\):
- Pair A: \(d = 5\), \(C = 22\).
The ratio is \(\frac{22}{5} = 4.4\).
Comparing this to \(\pi \approx 3.14\), the difference is \(|4.4 - 3.14| = 1.26\).
- Pair B: \(d = 19\), \(C = 60\).
The ratio is \(\frac{60}{19} \approx 3.15789\).
Comparing this to \(\pi \approx 3.14\), the difference is approximately \(|3.15789 - 3.14159| \approx 0.0163\).
- Pair C: \(d = 33\), \(C = 80\).
The ratio is \(\frac{80}{33} \approx 2.42424\).
Comparing this to \(\pi \approx 3.14\), the difference is approximately \(|2.42424 - 3.14159| \approx 0.71735\).
- Conclusion on reasonable approximations.
The ratio for Pair B (\(\approx 3.158\)) is the closest to the actual value of \(\pi\) (\(\approx 3.142\)). Pair A and Pair C deviate significantly from \(\pi\). Therefore, Pair B is the only reasonable approximation among the options.
The final answer is \(\boxed{B}\).
Examples
Imagine you are building a circular sandbox in your backyard. You decide that the diameter of the sandbox should be 19 inches. If you want to know how much rope you need for the perimeter of the circular edge, you use the relationship \(C = \pi d \approx 3.14 \times 19 \approx 59.66\) inches. Knowing that 60 inches is a close approximation helps you estimate that you need about 60 inches of rope, making this mathematical check very useful for construction and design projects.