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 substitute the value of t into the given function N(t) = 34.68 + 23.88 ln(1.05t + 5.3) and solve for N(t).
a. The year 2005 corresponds to t = 1 (since t = 0 corresponds to the year 2004). Plugging in t = 1 into the function, we have:
N(1) = 34.68 + 23.88 ln(1.05(1) + 5.3)
= 34.68 + 23.88 ln(1.05 + 5.3)
≈ 34.68 + 23.88 ln(6.35)
≈ 34.68 + 23.88(1.853)
≈ 34.68 + 44.03
≈ 78.71
Therefore, the projected number of online households at the beginning of 2005 is approximately 78.71 million.
b. To determine how fast the projected number of online households is 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.
First, let's find the derivative of N(t):
N'(t) = 0 + 23.88 * (1 / (1.05t + 5.3)) * 1.05
Simplifying that expression, we get:
N'(t) = 23.88 * 1.05 / (1.05t + 5.3)
Now, substitute t = 1 into N'(t):
N'(1) = 23.88 * 1.05 / (1.05(1) + 5.3)
= 23.88 * 1.05 / (1.05 + 5.3)
≈ 23.88 * 1.05 / 6.35
≈ 4.14
Therefore, the projected number of online households was increasing at a rate of approximately 4.14 million households per year at the beginning of 2005.