MHB How does temperature affect the length of a steel measuring tape?

[Joe_1234]

Please help me

A certain steel tape is known to be 100000 feet long when the temperature of 70°F . When the tape is at a temperature of 10°F, what reading corresponds to a distance of 90000 ft? Coefficient of linear expansion of the tape is 5.833 x 10-6 per °F.

 

[HOI]

My first thought was "what in the world is a "reading"? Finally I realized- this is NOT just a "steel tape" but a steel measuring tape! Let "x" be the length of the tape at 10 degrees F. When the temperature increases from 10 degrees F to 70 degrees F, an increase of 60 degrees F, the tape will have increased by a factor of 1+ 5.833 x 10^(-6) x 60= 1+ 349.98 X 10^(-6)= 1+ 3.4998 x 10^(-4)= 1.00034998. Since the length after the expansion is 100,000 feet, we must have 1.00034998x= 100,000. So the length of the tape at 10 degrees F is x= 100000/1.0034998= 99651.24058. The reading, at 90,000 feet, at 10 degrees F is given by \frac{y}{90000}= \frac{100000}{99651.24058}. y= 10(99651.24058)/9= 110723.6 feet.