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 (t=1), we can substitute t=1 into the function N(t).
a. Let's calculate it step by step:
Step 1: Substitute t=1 into the function N(t):
N(1) = 34.68 + 23.88 ln(1.05(1) + 5.3)
Step 2: Calculate 1.05(1) + 5.3:
1.05(1) + 5.3 = 1.05 + 5.3 = 6.35
Step 3: Substitute the value of 6.35 into the function:
N(1) = 34.68 + 23.88 ln(6.35)
Step 4: Calculate the natural logarithm (ln) of 6.35:
ln(6.35) ≈ 1.854
Step 5: Substitute the value of ln(6.35) into the function:
N(1) = 34.68 + 23.88(1.854)
Step 6: Calculate 23.88(1.854):
23.88(1.854) ≈ 44.238
Step 7: Add 34.68 and 44.238 to get the projected number of online households at the beginning of 2005:
N(1) ≈ 34.68 + 44.238 ≈ 78.918
Therefore, the projected number of online households at the beginning of 2005 is approximately 78.918 million.
To calculate the rate at which 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.
b. Let's find the derivative step by step:
Step 1: Start with the function N(t):
N(t) = 34.68 + 23.88 ln(1.05t + 5.3)
Step 2: Take the derivative of N(t) with respect to t:
N'(t) = 0 + 23.88 * d/dt (ln(1.05t + 5.3))
Step 3: Apply the chain rule to find the derivative of the natural logarithm:
N'(t) = 23.88 * (1/(1.05t + 5.3)) * d/dt (1.05t + 5.3)
Step 4: Take the derivative of 1.05t + 5.3 with respect to t:
N'(t) = 23.88 * (1/(1.05t + 5.3)) * 1.05
Step 5: Simplify the expression:
N'(t) = 23.88 * 1.05 / (1.05t + 5.3)
Therefore, the rate at which the projected number of online households was increasing at the beginning of 2005 (t=1) is given by N'(1) = 23.88 * 1.05 / (1.05(1) + 5.3).
To find the numerical value, we can substitute t=1 into the derivative equation:
N'(1) = 23.88 * 1.05 / (1.05 + 5.3)
N'(1) = 24.851 / 6.35 ≈ 3.91
Therefore, the projected number of online households was increasing at a rate of approximately 3.91 million households per year at the beginning of 2005.