Solving the Wheel Revolution Distance Problem

This question asks us to find out how many times a wheel with a specific diameter needs to spin completely to cover a given total distance. To solve this, we need to understand the relationship between a wheel's revolution and the distance it covers.

Understanding Wheel Revolutions and Distance

When a wheel makes one complete revolution, the distance it covers is equal to its circumference. The circumference of a circle (or a wheel) is calculated using the formula:

Circumference = $\pi \times \text{Diameter}$

where $\pi$ (pi) is a mathematical constant approximately equal to 3.14159 or $22/7$.

Calculating the Circumference of the Wheel

The diameter of the wheel is given as 84 cm. We can use the value $\pi = 22/7$ for this calculation as it simplifies the process with the given diameter.

• Diameter = 84 cm
• $\pi = 22/7$
• Circumference = $(22/7) \times 84 \text{ cm}$
• Circumference = $22 \times (84/7) \text{ cm}$
• Circumference = $22 \times 12 \text{ cm}$
• Circumference = 264 cm

So, in one complete revolution, the wheel covers a distance of 264 cm.

Converting Units for Consistency

The total distance to be covered is given in meters (792 m), while the circumference is in centimeters (264 cm). To perform calculations, both units must be the same. We will convert the total distance from meters to centimeters.

• 1 meter = 100 centimeters
• Total distance = 792 m
• Total distance in cm = $792 \times 100 \text{ cm}$
• Total distance in cm = 79200 cm

Calculating the Number of Complete Revolutions

The number of complete revolutions is found by dividing the total distance to be covered by the distance covered in one revolution (the circumference).

Number of Revolutions = $\frac{\text{Total Distance}}{\text{Circumference}}$

• Total Distance = 79200 cm
• Circumference = 264 cm
• Number of Revolutions = $\frac{79200 \text{ cm}}{264 \text{ cm}}$

Now, we perform the division:

• $79200 \div 264$
• Let's simplify the division: $792 \div 264$. We can see that $264 \times 3 = 792$.
• So, $79200 \div 264 = 300$.

Thus, the wheel will make 300 complete revolutions to cover a distance of 792 meters.

Final Answer on Wheel Revolutions

Based on our calculation, the number of complete revolutions the wheel will take to cover 792 m is 300.

Revision Table: Key Concepts for Wheel Problems

Additional Information on Circle Calculations

Here are some related concepts that are useful when dealing with problems involving circles and wheels:

• Value of $\pi$: The value of $\pi$ is an irrational number, meaning its decimal representation goes on forever without repeating. Common approximations are 3.14, 3.14159, or the fraction $22/7$. The choice of approximation often depends on the numbers involved in the calculation to make it easier.
• Area of a Circle: While not needed for this specific revolution problem, the area of a circle is another important concept, calculated as $A = \pi r^2$, where $r$ is the radius.
• Units: Always pay close attention to the units given in a problem (like cm and m). Ensure all values are in the same units before performing calculations. Unit conversion is a critical step in many physics and math problems.
• Complete Revolutions: The question asks for "complete revolutions," which implies we are interested in whole number spins. If the calculation resulted in a decimal, we might need to consider what "complete revolutions" means in context (e.g., rounding down). In this case, the result was a whole number.