The radius of a wheel is 14 m. How many kilometers will it travel if it turns 500 times?
Answer (2)
Calculate the circumference of the wheel: C = 2 π r = 2 π ( 14 ) = 28 π meters.
Determine the total distance traveled in meters: D i s t an ce = 28 π × 1000 = 28000 π meters.
Convert the total distance to kilometers: D i s t an ce = 1000 28000 π = 28 π kilometers.
The wheel travels approximately 87.96 kilometers.
Explanation
Problem Analysis The problem states that a wheel has a radius of 14 meters and turns 1000 times. We need to find the total distance the wheel travels in kilometers.
Calculate the Circumference First, we need to calculate the circumference of the wheel. The formula for the circumference of a circle (which represents the wheel) is given by: C = 2 π r where C is the circumference and r is the radius. In this case, the radius r = 14 meters.
Circumference Calculation Substituting the value of the radius into the formula, we get: C = 2 π ( 14 ) = 28 π So, the circumference of the wheel is 28 π meters.
Total Distance in Meters Next, we need to find the total distance traveled by the wheel after 1000 turns. The distance covered in one turn is equal to the circumference of the wheel. Therefore, the total distance in meters is: D i s t an ce = C i rc u m f ere n ce × N u mb ero f T u r n s D i s t an ce = 28 π × 1000 = 28000 π ≈ 87964.59 m e t ers
Convert to Kilometers Finally, we need to convert the distance from meters to kilometers. Since 1 kilometer = 1000 meters, we divide the total distance in meters by 1000: D i s t an ce in K i l o m e t ers = 1000 D i s t an ce in M e t ers D i s t an ce in K i l o m e t ers = 1000 28000 π = 28 π ≈ 87.96 ki l o m e t ers
Final Answer Therefore, the wheel will travel approximately 87.96 kilometers after 1000 turns.
Examples
Wheels are fundamental in transportation and machinery. Understanding how many rotations a wheel makes relative to the distance traveled is crucial in designing vehicles, calculating gear ratios, and even estimating wear and tear on tires. For instance, knowing the circumference of a car's tire helps determine how many times the tire rotates during a commute, which can then be used to estimate tire lifespan and fuel efficiency.
The wheel with a radius of 14 meters travels approximately 43.98 kilometers after turning 500 times. This calculation involves finding the circumference of the wheel and then multiplying by the number of turns. Finally, the distance is converted from meters to kilometers.
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