To determine the exponential growth equation, we can use the given data points and the concept of exponential growth.
Let's assume that the number of subscribers in Year 2007 represents the initial value, which we'll call "a." The exponential growth equation can be written as:
N = a * e^(kt)
Where:
- N represents the number of subscribers in a given year (t)
- e is the base of the natural logarithm, approximately 2.71828
- k is the growth rate constant
To find the growth rate constant (k), we can take the ratio of any two adjacent values of N and solve for k using the formula:
k = ln(N2/N1) / (t2 - t1)
where N2 and N1 are the number of subscribers in two adjacent years, and t2 and t1 are the corresponding years.
Let's calculate the growth rate constant using the data for Year 2007 and Year 2008:
k = ln(2.7/1.6) / (2008 - 2007)
≈ ln(1.6875) / 1
≈ 0.522618
Now that we have the value of k, we can calculate the exponential growth equation. Using the value of a = 1.6 million subscribers in Year 2007, the equation becomes:
N = 1.6 * e^(0.522618t)
To answer part A of your question, the exponential equation we found is:
N = 1.6 * e^(0.522618t)
Now let's move on to part B to determine during what year the number of subscribers will pass 100,000,000.
We need to solve the equation for t when N = 100,000,000:
100,000,000 = 1.6 * e^(0.522618t)
To isolate t, divide both sides of the equation by 1.6 and then take the natural logarithm of both sides:
ln(100,000,000 / 1.6) = 0.522618t
Now, divide the natural logarithm result by 0.522618 to solve for t:
t = ln(100,000,000 / 1.6) / 0.522618
Using a calculator, we can find that t ≈ 21.33 (rounded to two decimal places).
Since 't' represents the number of years from the initial year (2007), we can add 21.33 years to 2007 to determine the year when the number of subscribers will pass 100,000,000.
Thus, the number of subscribers is expected to surpass 100,000,000 in the year 2028 or early 2029.
Note: Keep in mind that this is an estimate based on the given data and assumes that the exponential growth trend will continue.