The number of U.S. households with ipods was estimated to be 55.5million in 2005

Question

The number of U.S. households with ipods was estimated to be 55.5million in 2005 & 70.3 million in 2007. Assume the number ofhouseholds with ipods grows exponentially.
a.) find the function of the number of U.S. households withipods t years after 2005.
b.) estimate the number of U.S. households that will haveipods in 2020
c.) In what year will the number of U.S. households with ipodsreach 100 million?
a.) find the function of the number of U.S. households withipods t years after 2005.
b.) estimate the number of U.S. households that will haveipods in 2020
c.) In what year will the number of U.S. households with ipodsreach 100 million?

Explanation / Answer

In this question we're assuming an exponential function (i.e. 'e'to the power of something) and as values change with time thisfunction is also a function of time (i.e. 'e' to the power ofsomething involving time). More precisely it will be "'e' to thepower of a constant, multiplied by the variable time, plus anotherconstant, equals number of ipods. Or... iPods = e(constant * time) + anotherconstant i = ext + c So generate two simultaneousequations. We're starting counting from 2005 so at that timet = 0 and from there we're counting in years so at 2007 t =2... In 2005       55.5 = e(x*0) +c   = ec In 2007       70.3 =e2x + c    and using rules of powers thisequals   70.3 = e2xec So the first equation's easy to solve ln(55.5) =ln(ec)                                                         4.02       = c Next substitute that back into 2nd equation... 70.3 = e2xe4.02 e2x = (70.3 / e4.02) ln(e2x) = ln(70.3 / e4.02) 2x = 0.232772 x = 0.12 So substitute x and c back into the generic equation and youhave... i = e0.12t +4.02 Hope that helped

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