The number of online buyers in Western Europe is expected to grow steadily in the coming years. The function below for 1 t 9, gives the estimated buyers as a percent of the total population, where t is measured in years, with t = 1 corresponding to 2001.
P(t) = 27.6 + 14.1 ln(t)
(a) What was the percent of online buyers in 2001 (t = 1)?
%
How fast was it changing in 2001?
%/yr
(b) What is the percent of online buyers expected to be in 2008 (t = 8)?
%
How fast is it expected to be changing in 2008?
%/yr
To find the percent of online buyers in 2001 (t = 1), we need to substitute t = 1 into the function P(t) = 27.6 + 14.1 ln(t):
P(1) = 27.6 + 14.1 ln(1)
Since the natural logarithm of 1 is 0, we can simplify the equation:
P(1) = 27.6 + 14.1 * 0
P(1) = 27.6
Therefore, the percent of online buyers in 2001 is 27.6%.
To find how fast it was changing in 2001, we can differentiate the function P(t) with respect to t and then substitute t = 1 into the derivative:
P'(t) = dP/dt = 14.1 / t
P'(1) = 14.1 / 1
P'(1) = 14.1
Therefore, it was changing at a rate of 14.1%/yr in 2001.
To find the percent of online buyers expected to be in 2008 (t = 8), we need to substitute t = 8 into the function P(t):
P(8) = 27.6 + 14.1 ln(8)
Using a calculator, ln(8) ≈ 2.0794, so we can calculate the result:
P(8) ≈ 27.6 + 14.1 * 2.0794
P(8) ≈ 27.6 + 29.34454
P(8) ≈ 56.94454
Therefore, the percent of online buyers expected to be in 2008 is approximately 56.94%.
To find how fast it is expected to be changing in 2008, we can differentiate the function P(t) with respect to t and then substitute t = 8 into the derivative:
P'(t) = dP/dt = 14.1 / t
P'(8) = 14.1 / 8
P'(8) ≈ 1.7625
Therefore, it is expected to be changing at a rate of approximately 1.7625%/yr in 2008.
(a) To find the percent of online buyers in 2001 (t = 1), we need to substitute t = 1 in the given function:
P(1) = 27.6 + 14.1 ln(1)
Since ln(1) = 0, the equation becomes:
P(1) = 27.6 + 14.1(0)
P(1) = 27.6
Therefore, the percent of online buyers in 2001 is 27.6%.
To find how fast it was changing in 2001, we need to find the derivative of the function with respect to t and then substitute t = 1:
P'(t) = 14.1/t
P'(1) = 14.1/1
P'(1) = 14.1
Therefore, it was changing at a rate of 14.1%/yr in 2001.
(b) To find the percent of online buyers expected to be in 2008 (t = 8), we need to substitute t = 8 in the given function:
P(8) = 27.6 + 14.1 ln(8)
Since ln(8) is approximately 2.08, the equation becomes:
P(8) = 27.6 + 14.1(2.08)
P(8) = 27.6 + 29.328
P(8) = 56.928
Therefore, the percent of online buyers expected to be in 2008 is approximately 56.928%.
To find how fast it is expected to be changing in 2008, we need to find the derivative of the function with respect to t and then substitute t = 8:
P'(t) = 14.1/t
P'(8) = 14.1/8
P'(8) = 1.7625
Therefore, it is expected to be changing at a rate of 1.7625%/yr in 2008.