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In the problem below I don't understand the calculation in which the coefficient is multiplied by 60 degrees celsius. How is this done? can someone expand that calculation into simpler terms.

In short I guess I don't understand the relations with the coefficient and the degrees.

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The difference in temperature is -25-35 = -60. That's multiplied to the coefficient of linear expansion and the original length to find the change in length. Coefficient of linear expansion in other words is the linear expansion of a specific material of unit length, per unit change in temperature, which is Celsius in your case.

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