Revolutions Calculation for Wheel Travel

The problem asks us to find out how many times a wheel of a specific diameter rotates (makes revolutions) while traveling a given distance. The key relationship here is that the distance covered by the wheel in one full revolution is equal to its circumference.

Understanding Circumference and Revolutions

The circumference of a circle is the distance around its outer edge. For a wheel, this is the length of the path its edge traces on the ground in one full turn. If a wheel makes 'N' revolutions, the total distance traveled is N times the circumference of the wheel.

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

\(C = \pi \times d\)

where \(d\) is the diameter of the circle, and \(\pi\) (pi) is a mathematical constant, approximately equal to \(3.14159\) or often taken as \(\frac{22}{7}\) for calculations involving multiples of 7.

Given Information

• Diameter of the wheel (\(d\)) = 56 cm
• Total distance traveled = 22 km

Unit Conversion

To perform calculations, we must ensure all measurements are in the same units. Let's convert the distance from kilometers to centimeters.

• 1 km = 1000 meters
• 1 meter = 100 centimeters
• So, 1 km = 1000 \(\times\) 100 cm = 100,000 cm

Total distance traveled = 22 km = 22 \(\times\) 100,000 cm = 2,200,000 cm.

Calculating the Circumference

Using the diameter \(d = 56\) cm and taking \(\pi = \frac{22}{7}\):

\(C = \pi \times d\)

\(C = \frac{22}{7} \times 56 \text{ cm}\)

\(C = 22 \times \frac{56}{7} \text{ cm}\)

\(C = 22 \times 8 \text{ cm}\)

\(C = 176 \text{ cm}\)

The circumference of the wheel is 176 cm.

Calculating the Number of Revolutions

The total distance traveled is the number of revolutions multiplied by the circumference:

Total Distance = Number of Revolutions \(\times\) Circumference

Let \(N\) be the number of revolutions. We have:

\(2,200,000 \text{ cm} = N \times 176 \text{ cm}\)

To find \(N\), we divide the total distance by the circumference:

\(N = \frac{\text{Total Distance}}{\text{Circumference}}\)

\(N = \frac{2,200,000 \text{ cm}}{176 \text{ cm}}\)

Now, let's perform the division:

\(N = \frac{2,200,000}{176}\)

\(N = \frac{2200000}{11 \times 16}\)

\(N = \frac{200000}{16}\)

\(N = \frac{100000}{8}\)

\(N = \frac{50000}{4}\)

\(N = 12500\)

The wheel makes approximately 12500 revolutions.

Final Answer Check

The calculated number of revolutions is 12500, which matches one of the given options.

Therefore, a wheel of 56 cm diameter will make approximately 12500 revolutions in traveling 22 km.