Suppose the number of DVDs shipped by DVD manufacturers in a region in 1997 was 200,000. In 2000, the number had increased to 3,700,000 If it is assumed that number of DVDs shipped by manufactures grows at an exponential rate, how many DVDs will be shipped in 2010. Round your answer to the nearest hundred thousand.
A)61,959,400,000
B)433,399,700,000
C)21,200,000
D)23,427,000,000
E)3,349,200,000

Question

Quizplus · Accepted Answer

We can use the exponential growth formula: N(t) = N0 * e^(rt), where N(t) is the number of DVDs shipped at time t, N0 is the number of DVDs shipped initially, r is the growth rate, and e is Euler's number (approximately 2.71828). 
Using the data given, we can find r: 
3,700,000 = 200,000 * e^(3r)
e^(3r) = 3,700,000/200,000 = 18.5
3r = ln(18.5) (taking natural logarithm on both sides)
r = ln(18.5)/3 ≈ 1.6536
So, the growth rate is approximately 165.36% per year. 
Using N(t) = N0 * e^(rt) again, we can find the number of DVDs shipped in 2010 (t = 13, since 2010 is 13 years after 1997): 
N(13) = 200,000 * e^(1.6536*13) ≈ 23,427,000,000
Rounding to the nearest hundred thousand, we get 23,427,000,000 as the answer, which is option D.

Suppose the Number of DVDs Shipped by DVD Manufacturers in a Region