You are on a tight budget so you’ve let your tires get more worn than they should and now their radius is 15 mm less than they were originally. Is your odometer over-estimating or under-estimating the distance you drive?
To determine whether your odometer is over-estimating or under-estimating the distance you drive, we need to understand the relationship between the tire radius and the distance measured by the odometer.
As tires spin, the odometer calculates the distance traveled based on the number of revolutions made by the wheels. The circumference of a tire can be calculated using the formula: C = 2πr, where C is the circumference and r is the radius.
Given that the tire radius is 15 mm less than it originally was, we can calculate the difference in circumference between the worn tires and the original tires using the formula: ΔC = 2π(Δr).
In this case, the difference in radius (Δr) is -15 mm, as it is decreased by 15 mm. Thus, the difference in circumference (ΔC) is:
ΔC = 2π(-15)
By evaluating this, we obtain the difference in circumference as -30π mm.
Now, to determine whether the odometer is over-estimating or under-estimating, we need to consider that the odometer measures the distance based on the tire circumference. If the tire circumference is smaller due to the reduced radius, then the tire travels a shorter distance per revolution compared to the original tires.
Since the odometer measures distances based on the assumption of the original tire circumference, it will over-estimate the distance traveled with the worn tires. Therefore, the odometer in this scenario is over-estimating the distance you drive.