A bicycle wheel is 63 centimeters from top to bottom. When the wheel goes all the way around one time, the bicycle travels 198 centimeters. How can this information be used to estimate the value of pi ?
The Correct Answer and Explanation is:
We are given two pieces of information about a bicycle wheel:
- The diameter of the wheel is 63 centimeters (from top to bottom).
- When the wheel rolls one full revolution, the distance it travels (its circumference) is 198 centimeters.
Step-by-Step Solution
We can use the formula for the circumference of a circle: C=π×dC = \pi \times d
Where:
- CC is the circumference (the distance the wheel travels in one revolution),
- dd is the diameter of the wheel,
- π\pi is the constant we want to estimate.
We are given:
- C=198C = 198 cm
- d=63d = 63 cm
Now substitute the values into the formula and solve for π\pi: 198=π×63198 = \pi \times 63
Divide both sides by 63: π=19863=227≈3.14\pi = \frac{198}{63} = \frac{22}{7} \approx 3.14
Final Answer:
π≈19863=227≈3.14\boxed{\pi \approx \frac{198}{63} = \frac{22}{7} \approx 3.14}
Explanation
This problem allows us to estimate the value of π (pi) using real-world measurements. Pi is a mathematical constant that represents the ratio of a circle’s circumference to its diameter. No matter the size of the circle, this ratio is always the same: π≈3.14159…\pi \approx 3.14159…
In the case of the bicycle wheel, we are told that its diameter (distance across the circle through its center) is 63 centimeters. When the wheel turns exactly one full time (a single revolution), the bicycle moves forward 198 centimeters. That forward distance is the same as the wheel’s circumference — the total distance around the circle.
Using the relationship circumference=π×diameter\text{circumference} = \pi \times \text{diameter}, we can rearrange the formula to solve for π\pi: π=circumferencediameter=19863\pi = \frac{\text{circumference}}{\text{diameter}} = \frac{198}{63}
When we divide 198 by 63, we get approximately 3.14, which is a close approximation of pi. Interestingly, 19863\frac{198}{63} simplifies to 227\frac{22}{7}, which is a well-known fraction used to estimate pi, especially in basic geometry.
This problem shows how pi is not just an abstract number but is also measurable in real life. By observing how far a circular object travels in one full rotation and knowing its diameter, we can estimate pi using simple division.
