The amount of freight transported by rail in the United States was about 580 billion ton-miles in 1960 and has been increasing at a rate of 2.32% per year since then. (a) Write a function representing the amount of freight, in billions of ton-miles, transported annually (1960 = year 0). (b) Graph the function. (c) In what year would you predict that the number of ton-miles would have exceeded or would exceed 1 trillion (1000 billion)?
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2.32% = 0.0232
f(t) = 580(1+0.0232)^t
You want 580 x 10^9 (1.0232)^N to be one trillion, i.e. 10^12.
Thus 580 (1.0232)^N = 10^3
1.0232^N = 1000 / 580 = 1. 17241
Take logs of both sides.
N log(1.0232) = log (1.17241)
N = 0.06944/0.00995
N = 6.979 (years)
So it needs 7 full years from 1960 to reach one trillion.
So it occurs in 1967