(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.

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.