How many revolutions approximately will a wheel of 56 cm diameter make in traveling 22 km?
12500
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.
Important Questions from Plane Figures
If length of a rectangle is increased to its three times and breadth is decreased to its half, then the ratio of the area of given rectangle to the area of new rectangle is:
The width of the path around a square field is 4.5 m and its area is 105.75 m 2. Find the cost of fencing the field at the rate of Rs. 100 per meter.
What is the area of the square (in cm 2) whose vertices lie on a circle of radius 5 cm?
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
The perimeter of a circular lawn is 1232 m. There is 7 m wide path around the lawn. The area (in m 2) of the path is:
Take \(\left(\pi=\frac{22}{7}\right)\)
