# 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 ------? decreasing constant increasing 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. (c) According to this model, in what year did the number of hosts reach 49 million?

Question
Exponential models
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 ------? decreasing constant increasing 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.
(c) According to this model, in what year did the number of hosts reach 49 million?

2020-11-09
(a) Given that between 1995 and 2005, each year the number of Internet domain hosts was 1.43 times the number of hosts in the preceding year.
This implies that if $$N_0$$ is the number of hosts in the initial year, then the number of hosts $$N_1$$ in the next year is given by $$N_1=1.43N_0$$
Using the number of hosts $$N_1$$ as the number in the previous year, the number of hosts in the next year $$N_2$$ is given by $$N_2=1.43N_1=1.43(1.43N_0)=(1.43)^2N_0$$
Using the number of hosts $$N_2$$ as the number in the previous year, the number of hosts in the next year $$N_3$$ is given by $$N_3=1.43N_2=1.43((1.43)^2N_0)=(1.43)^3N_0$$
Generalising the above equations, we get $$N_t=(1.43)^tN_0$$
Let N represents the number of hosts after t years from 1995.
Then, we can replace $$N_t$$ by N in $$N_t=(1.43)^tN_0$$
This gives the equation $$N=(1.43)^tN_0$$
The above equation can be compared to the general form of an exponential function $$y=ab^x$$ where $$y\approx N,x\approx t,N_0\approx a,b\approx1.43$$
So, the number of hosts is an exponential function of time.
The number of hosts grows by a factor of 1.43 each year, this is an exponential function because the number is growing by 1.43 increasing multiples.
(b) Let N be the number of hosts in millions, as a function of the time t in years since 1995.
As seen in part (a), we have the equation $$N=(1.43)^tN_0$$ where $$N_0$$ represents the initial number of hosts.
Given that the initial number of hosts is 8.2 million.
Substitute the value $$N_0=8.2$$ in $$N=(1.43)^tN_0$$ and obtain $$N=(1.43)^t\times8.2=8.2(1.43)^t$$
Therefore, the formula for the number of hosts in millions as a function t in years since 1995 is $$N=8.2(1.43)^t$$
c) Substitute the value N = 49 in the formula $$N=8.2(1.43)^t$$ to identify the year in which the number of hosts reached 49 million.
$$49=8.2(1.43)^t$$
$$\Rightarrow(1.43)^t=\frac{49}{8.2}$$
$$\Rightarrow\ln(1.43)^t=\ln(\frac{49}{8.2})$$
$$\Rightarrow t\ln(1.43)=\ln(\frac{49}{8.2})$$
$$\Rightarrow t=\frac{\ln(\frac{49}{8.2})}{\ln(1.43)}$$
$$\Rightarrow t\approx4.99$$
$$\Rightarrow t\approx5$$
So, it took 5 years since 1995 for the number of hosts to be 49 million.
Therefore, the number of hosts reached 49 million in the year 2000.

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