1.1) #36 For small changes in temperature, the formula for the expansion of a metal rod under a change in temperature is: l-lo = alo(t-to), where l is the length of the object at temperature t, and lo is the length at temperature to, and a is a constant which depends on the type of metal. a) Express l as a linear function of t. Find the slope and vertical intercept in terms of lo, to, and a. b) A rod is 100cm long at 60o F and made of a metal with a = 10-5. Write an equation giving the length of this rod at temperature t. c) What does the sign of the slope tell you about the expansion of a metal under a change in temperature? a) l is found, as a linear function of t, by getting the equation into an (l =) format. To do this, since we already have l - lo on the left side, we simply add lo to both sides, to get l alone on the left side and giving us our (l =) equation we were looking for. l-lo + lo = alo(t-to) + lo The two los on the left side cancel out, giving us l = alo(t-to) + lo And to get the equation into the linear, y = mx + b form, we distribute the lo, into (t-to), so we can have the constant, a, alone on the outside of the parenthesis, and this gives us l = a(lot - tolo) + lo , which is now in slope intercept form, l being the equivalent of y, a is the equivalent of m, (lot - tolo) is the equivalent of x, and lo is the equivalent of b. The slope, or constant, which in the slope intercept equation (y = mx + b) is represented by m, which we have already established is equivalent to the a in our equation for finding the length of a rod, is found by taking the equation that we have above, l = a(lot - tolo) + lo, and solving for the slope, a. To do this, simply subtract both sides by lo, then divide both sides by (lot - tolo), to get a by itself, therefor solving for a, on the right side. l - lo = a(lot - tolo) + lo - lo The lo on the right side cancels out, leaving l - lo = a(lot - tolo)
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