Often new technology spreads exponentially. Between 1995 and 2005, each year the number of Internet domain hosts was 1.43 times the number of hosts in the preceding year. In 1995, the number of hosts was 8.2 million. (a) Explain why the number of hosts is an exponential function of time. The number of hosts grows by a factor of __________ each year, this is an exponential function because the number is growing by constant multiples. (b) Find a formula for the exponential function that gives the number N of hosts, in millions, as a function of the time t in years since 1995. N= (c) According to this model, in what year did the number of hosts reach 35 million? (Round your answer to the nearest year.)

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Often new technology spreads exponentially. Between 1995 and 2005, each year the number of Internet domain hosts was 1.43 times the number of hosts in the preceding year. In 1995, the number of hosts was 8.2 million.  (a) Explain why the number of hosts is an exponential function of time. The number of hosts grows by a factor of __________ each year, this is an exponential function because the number is growing by  constant multiples.  (b) Find a formula for the exponential function that gives the number N of hosts, in millions, as a function of the time t in years since 1995.  N=  (c) According to this model, in what year did the number of hosts reach 35 million? (Round your answer to the nearest year.)

Velsenw · Accepted Answer

Step 1  a) For an exponential function, a is the base, P is the initial value and N denotes the number at time t.  After 1 year, the number of internet hosts becomes,  N(1)=P(1.43)1  =1.43P  After 2 years,  N(2)=P(1.43)2  =2.045P  The numbers of hosts grow by a factor of 1.43 each year; this is an exponential function because the number is growing by 1.43 multiples.  Step 2  b) To determine a formula for the exponential function that gives the number N of hosts, in millions, as a function of the time t in years since 1995.  Given that, the number of hosts was 8.2 million in 1995.  Step 3  The initial number of hosts is,  N(0)=8.2 million  The number of hosts after 1 year,  N(1)=(1.43)N(0)  =(1.43)(8.2)  The number of hosts after 2 years,  N(2)=(1.43)N(1)  =(1.43)((1.43)(8.2))  =(1.43)2(8.2)  The number of hosts after 3 years,  N(3)=(1.43)N(2)  =(1.43)((1.43)2(8.2))  =(1.43)3(8.2)  Therefore, the number of hosts, in millions, as a function of the time t in years since 1995 is,  N(t)=(1.43)t(8.2)  Step 4  c) To determine the year in which the number of hosts reach 35 million.  Substitute N=35 in the above formula for exponential function. S N(t)=(1.43)t(8.2)  35=(1.43)t(8.2)  4.27=(1.43)t  ln⁡(4.27)=tln⁡(1.43)  t=4.05  t≈4 years  Thus, the number of hosts reaches 35 million in the year 1999.

Often new technology spreads exponentially. Between 1995 and 2005, each year the

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