The total number of DVD videos produced and shipped in 1998 was .5 million. In 2004,the total number of units reached 29.01 million.

Assuming the exponential model applies:
a)Find the value of k and write the function
b) Estimate the number of DVD videos produced and shipped in 2005,in2008,and in 2011

To find the value of k and write the exponential growth function, we can use the formula:

P = P0 * e^(kt)

where:
P0 = initial quantity (in this case 0.5 million)
P = final quantity (in this case 29.01 million)
t = time period (in this case 2004 - 1998 = 6)

a) We can now substitute the given values into the formula and solve for k:

29.01 = 0.5 * e^(6k)

Divide both sides of the equation by 0.5:

58.02 = e^(6k)

Taking the natural logarithm of both sides:

ln(58.02) = ln(e^(6k))

Using the property ln(e^x) = x:

ln(58.02) = 6k

Divide by 6:

k = ln(58.02) / 6

Using a calculator, we find k ≈ 0.2333.

b) To estimate the number of DVD videos produced and shipped in 2005, 2008, and 2011, we can use the exponential growth function:

P = P0 * e^(kt)

For 2005: t = 2005 - 1998 = 7
P(2005) = P0 * e^(k * 7)

For 2008: t = 2008 - 1998 = 10
P(2008) = P0 * e^(k * 10)

For 2011: t = 2011 - 1998 = 13
P(2011) = P0 * e^(k * 13)

Substituting the values of P0, k, and t into the equations, we can calculate the estimates for the number of DVD videos produced and shipped in each year.