Dear Tom and Ray: It occurred to me the other day that as a car is driven and the tires wear down, each tire's circumference decreases. Since the speedometer measures speed through the revolutions of the drive shaft or something, and then converts that to miles per hour, isn't it based on an assumption that your tires have a certain circumference? My question is, How much does tread wear affect the speedometer reading? Is it significant? And can I use this argument to get out of a speeding ticket? Thanks.---Todd

How would you respond, with sufficent mathematical reasoning as justification?

Source: "Car Talk," *Bellingham Herald*, October 15, 2010

**Hint:** Make some assumptions about tire size and tread change...and then start computing....

**Solution Commentary:** Here is Tom and Ray's response...Do you agree with their reasoning?

**Ray:** We'll do the math for you. Let's say a typical new tire is 25 inches in diameter. And let's say there's half an inch of tread all the way around the outside. Therefore, whenh the tire goes from new to old, you'd lose an inch in diameter (half an inch on either side). So, in our scenario, a typical bald tire is 24 inches in diameter.
**Tom:** we know that the circumference of a circle is directly proportional to its diameter. So we can conclude that when you go from 25 to 245 inches in diameter, the change in the speedometer reading is 4 percent.

**Ray:** So the difference between a brand-new tire and a baldie changes your speedometer reading by less than 3 mph when you're going 65.

**Tom:** That's probably not enough to change a speeding ticket, Todd. besides, it goes in the WRONG direction. As your tires shrink in size, the car is actually going slower than the speedometer reads, not faster. So while your speedometer read 80, your ticket only cited you for going 77. But you already knew that.