Question 1110910: Your car's speedometer is geared to accurately give your speed using a certain tire size: 15" diameter wheels (the metal part) and 4.3" tires (the rubber part).
(a) If your car's instruments are properly calibrated, how many times should your tire rotate per second if you are travelling at 55 mi/hr?
rotations =
Report answer accurate to 3 decimal places.
(b) You buy new 5.4" tires and drive at a constant speed of 65 mph (according to your car's instrument). However, a cop stops you and claims that you were speeding. How fast did the radar gun clock you moving?
actual speed = ?mph
Report answer accurate to the nearest whole number.
(c) Then you replace your tires with 3.45" tires. When your speedometer reads 30 mph, how fast are you really moving?
actual speed = ?mph
Report answer accurate to 1 decimal places.
Found 2 solutions by Theo, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the following chart shows the results of my analysis.
what the speedometer reports as the speed is determined by the following.
the diameter of the tire is 15 + 2 * 4.3 = 23.6 inches.
at 55 miles per hour, the number of rotations of the tire per second is 55 * 5280 * 12 / 3600 / (23.6 * pi) = 13.05610042.
the ratio of the miles per hour to number of rotations per second is 55 / 13.05610042 = 4.212590149.
as long as the diameter of the tire is 23.6 inches, that ratio will be constant.
at 55 miles per hour, the tires will be rotating 55 / 4.212590149 = 13.05610042 times per second.
at 65 miles per hour, the tires will be rotating 65 / 4.212590149 = 15.42993686 times per second.
at 30 miles per hour, the tires will be rotating 30 / 4.212590149 = 7.12509318 times per second.
the multiplication factor of 4.212590149 * the number of rotations of the tire per second is what the speedometer of the car is displaying.
this multiplication factor is shown in row 4 column F of the chart.
similar calculations are done for tire diameters of 25.8 inches and 21.9 inches.
tire diameter of 25.8 inches is determined by taking 15 inches + 2 * 5.4 inches.
tire diameter of 21.9 inches is determined by taking 15 inches + 2 * 3.45 inches.
the 15 inches is the diameter of the metal part of the tires.
2 times the 5.4 inches and 2 times the 3.45 inches are the amount of inches that the rubber part of the tire adds to the total diameter of the tire (metal part plus rubber part).
the multiplication factor for 25.8 inch diameter tire is 4.605289231 and is shown in row 5 column F of the chart.
the multiplication factor for 21.9 inch diameter tire is 3.909140859 and is shown in row 6 column F of the chart.
when the diameter of the tire is 23.6 inches and the number of rotations of the tire per second is 15.42993686 times per second, the actual speed of the car is 65 miles per hour.
this is shown in row 4 column H of the chart.
this is the same as what the speedometer of the car shows, which is to be expected, since the speedometer is calibrated to a tire with a diameter of 23.6 inches.
when the diameter of the tire is 25.8 inches and the number of rotations of the tire is 15.42993686 times per second, the actual speed of the car is 71.05932203 miles per hour.
this is shown in row 5 column H of the chart.
this is the speed that the cop radar shows.
the driver thought he was driving at 65 miles per hour, but he was actually driving at 71.05932203 miles per hour because the diameter of the tires is larger than what the speedometer was calibrated at.
when the diameter of the tires is 21.9 inches and the number of rotations of the tire is 7.121509318 times per second, the actual speed of the car is 27.83898305 miles per hour.
this is shown in row 6 column I of the chart.
the driver thought he was driving at 30 miles per hour, but he was actually driving at 27.83898305 miles per hour because the diameter of the tires is smaller than what the speedometer was calibrated at.
your solutions to the following questions are therefore:
(a) If your car's instruments are properly calibrated, how many times should your tire rotate per second if you are travelling at 55 mi/hr?
Report answer accurate to 3 decimal places.
rotations = 13.056 times per second.
(b) You buy new 5.4" tires and drive at a constant speed of 65 mph (according to your car's instrument). However, a cop stops you and claims that you were speeding. How fast did the radar gun clock you moving?
Report answer accurate to the nearest whole number.
actual speed = 71 miles per hour.
(c) Then you replace your tires with 3.45" tires. When your speedometer reads 30 mph, how fast are you really moving?
Report answer accurate to 1 decimal places.
actual speed = 27.8 miles per hour.
i'm reasonably sure this is a correct analysis.
let me know if you have any further questions regarding it.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
I didn't check all of tutor theo's analysis, but his process seems right and so his answers should be good.
It is easy to show that his answers for parts b and c are right.
A speedometer determines the car's speed by counting the revolutions of the axle. It shows the correct speed if the tires being used are the ones for which the speedometer was calibrated.
If you put on larger tires, the circumference of the tire is larger, so you go farther on each revolution, so your speed is higher than what the speedometer shows; if you put on smaller tires, you speed is lower than what the speedometer shows.
Since the ratio of the circumferences of two different size tires is the same as the ratio of the diameters, the ratio of actual speed to the speed shown by the speedometer is just the ratio of the diameters.
For part b of the question then...
The diameter of the wheel for which the speedometer is calibrated is 15+2*4.3 = 23.6 inches. When you put on 5.4 inch tires, the diameter is 15+2*5.4 = 25.8 inches.
If the speedometer shows 65 mph when you are using the larger tires, your actual speed is 65 * (25.8/23.6) = 71.06 mph.
For part c, the new diameter of the wheel is 15+2*3.45 = 21.9 inches. So when the speedometer reads 30 mph with the smaller tires, your actual speed is 30 *(21.9/23.6) = 27.84 mph.
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