Cash Social Security distributions were $35 million, or $0.035 billion, in 1940.

Question

Cash Social Security distributions were $35 million, or $0.035 billion, in 1940. The amount has increased exponentially to $492 billion in 2004. Assuming that the exponential growth model applies: a.) Find the exponential growth rate k. b.) Find the exponential growth function. c.) Estimate the cash distributions in 1965 and in 2015. d.) In what year will the cash benefits reach $1 trillion? The work and solutions are on the textbook solutions, however I am unable to read the work because it shows up in 12 point font. I would GREATLY appreciate it if someone could help me especially since I have my math final Tuesday morning...Thanks!!

Explanation / Answer

The exponential model : C(t) = Ce^kt

at t= 0 ; C = $0.035 billion

at t= 64yrs ; C = 492 billion

Plug these value to find k:

492 = 0.035 e^(64k)

take natural log on both sides:

ln(492/0.035) = 64k

a) Growth factor , k = 0.149

b) Growth function:

C(t) = 0.035e^(0.149t)

c) in 1995 t = 55yrs

C(55) = 0.035e^(0.149*55) = 0.035*3640.95 = 127.43 billion

in 2015 ; t = 75 yrs

C(75) = 0.035e^(0.149*75 ) = 0.035*71324.84 = $ 2496.36 billion

d) In what C(t) = 1 trillion = 1000 billion

So, 1000 = 0.035e^(0.149t)

taking natural log on both sides:

ln(1000/0.035) = 0.149t

t = 68.86 years i.e. year 2009

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