A popular search engine is targeting European countries where the number of online households is expected to grow at a steady rate. Data was taken over time, and it was found that the number of online houses (in millions) projected can be modeled by the following function:
N(t) = 34.68 + 23.88 ln(1.05t + 5.3), f or 0 _< t _< 2.
The values of t are in years and when t = 0 , the year is 2004. Address the following questions:
a. What was the projected number of online households at the beginning of 2005?
b. How fast was the projected number of online households increasing at the beginning of 2005
Please i need step by step details so i will know where i am having problem on the solution. Thanks
To find the projected number of online households at the beginning of 2005, we need to find the value of N(t) when t = 1, because t represents the number of years since 2004.
Step 1: Substitute t = 1 into the equation N(t) = 34.68 + 23.88 ln(1.05t + 5.3):
N(1) = 34.68 + 23.88 ln(1.05(1) + 5.3)
N(1) = 34.68 + 23.88 ln(1.05 + 5.3)
N(1) = 34.68 + 23.88 ln(6.35)
Step 2: Use a calculator to evaluate the natural logarithm (ln) of 6.35:
ln(6.35) ≈ 1.850
Step 3: Substitute the result into the equation:
N(1) ≈ 34.68 + 23.88 (1.850)
N(1) ≈ 34.68 + 44.748
N(1) ≈ 79.428
Therefore, the projected number of online households at the beginning of 2005 is approximately 79.428 million (rounded to three decimal places).
To find how fast the projected number of online households was increasing at the beginning of 2005, we need to find the derivative of the function N(t) with respect to t and evaluate it at t = 1.
Step 1: Differentiate the function N(t) = 34.68 + 23.88 ln(1.05t + 5.3) with respect to t:
N'(t) = 23.88 * (1.05t + 5.3)^(-1) * 1.05
Step 2: Substitute t = 1 into the derivative equation:
N'(1) = 23.88 * (1.05(1) + 5.3)^(-1) * 1.05
N'(1) = 23.88 * (1.05 + 5.3)^(-1) * 1.05
N'(1) = 23.88 * (6.35)^(-1) * 1.05
Step 3: Use a calculator to evaluate (6.35)^(-1):
(6.35)^(-1) ≈ 0.157
Step 4: Substitute the result into the derivative equation:
N'(1) ≈ 23.88 * 0.157 * 1.05
N'(1) ≈ 3.737
Therefore, the projected number of online households was increasing at a rate of approximately 3.737 million households per year at the beginning of 2005.