W. Courtney Trabue
EMT-725 Problem Solving
Dr. Jim Wilson (Spring 1997)

Big Tire Problem

Problem Statement: A Bronco was originally fitted with 30 inch tires. Thus, the speedometer is calibrated for 30 inch diameter tires. Then, 31 inch diameter tires were put on the Bronco.

How did this change the speedometer readings?

What should the driver make the speedometer read (with the 31 inch tires) so that the actual speed is 55 mph?

Solution:

Compare the "smaller tire truck" with the "big tire truck".

Let

Ds = distance traveled by the smaller tire truck
ds = diameter of smaller tire ( 30 inches )

Db = distance traveled by the big truck.
db = diameter of big tire ( 31 inches )

Assume Ds = Db and convert to revolutions.

Ds [1 rev / ( (pi) ds)] = Db [ X rev/ ( (pi) db)]

Now, we know by algebra that

1 / ds = X / db

So

X = db / ds

Now let

ts = revolutions of the smaller tires
tb= revolutions of the bigger tires

Then

ts = X ( tb ) for the same distance traveled.

For the smaller tires, S ( ts ) = speedometer reading,
where S is some constant (determined by tire size ) that gives a mph reading
when multiplied by the number of tire revolutions (ts in this case).

So we're interested in S (ts) = 55 mph.

By algebra we get ts = 55 / S.

For the bigger tires,
we have S ( tb ) = x, where x is the speedometer reading.

Because tb = ts / X = ( ts ) ( ds / db ),

by substituting in for tb we get:

S ( ts ( ds / db ) ) = x.

So

S [ ( 55 / S ) ( ds / db) ] = x.

Therefore, 55 ( ds / db ) = x generates 55 (0.9677) = x.

The speedometer reading for an actual speed of 55 mph is x = 53.2258.
Using Excel Spread Sheets and the equation s ( ds / db ) = x generates several speedometer readings for actual speeds s.



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