Paper Topic:

Tire Testing

TIRE TESTING

Tire Testing

TIRE TESTING Page 1 of 2

The circumference of the tire that fits on a 16 inch wheel is greater than the circumference of a tire on a 15 inch wheel . Discounting sidewall depth , which would be the same on both tires since they differ only by diameter , the circumference (2 Pi radius ) of the 15 inch tire is 47 .12 ' and the circumference of the 16 inch tire is 50 .27 2000 miles equals 10 ,560 ,000 feet , which equals 126 ,720 ,000 inches Therefore the

15 inch tire makes (0 /47 .12 ) 2 ,689 ,303 revolutions in 2000 miles , and the 16 inch tire makes (0 /50 .27 ) 2 ,520 ,787 revolutions . Since the 16 inch tire makes fewer revolutions , the tire tread wears less and therefore the car with the 15 inch tires will need new tires first

2 ) The general form of the tangent sum formula is Tan (X Y (Tan x Tan Y (1- Tan

X Tan Y . The tangent function approaches infinity as the angle approaches 90 degrees

270 degrees , 450 degrees , etc , or in other words , Pi /2 radians Therefore Tan (X 450 (Tan X Tan 450 (1-Tan X Tan 450 . But Tan 450 is undefined , and therefore this

solution is undefined . However , using sine and cosine functions Tan (x 450 Sin (x

450 / Cos (X 450 . Since 450 degrees equals Pi /2 radians , and 90 degrees equals Pi /2

radians , this reduces to Sin (X 90 / Cos (X 90 . This in turn equals (Sin X Cos 90

Sin 90 Cos X (Cos X Cos 90 - Sin X Sin 90 . Of course , Sin 90 1 and Cos 90 0

Therefore this equals (Sin X (0 (1 ) Cos X (Cos X (0 ) - Sin X (1 ) which equals

Cos X / -Sin X which equals -Cot X (Zwillinger , Krantz , and Rosen 1996 , pp . 447-472

TIRE TESTING Page 2 of 2

A trig identity that is an equation that is true for all angles . An example of a trig identity is Sin X Tan X Cos X . This is true for all angles X . A trig equation that is not an identity is only true for some angles . Using the example above , Cos X 0 is only valid for angles X that are Pi /2 radians , or 90 degrees , 270 degrees , 450 degrees , etc (Zwillinger , Krantz , and Rosen 1996 , pp . 447 - 472

References

Zwillinger , D , Krantz , S , Rosen , K (Eds (1996 . Standard Mathematical Tables and

Formulae . New York : CRC Press...

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