In 1995, there were 5,400 cell phone users in the city of Northridge. The following year, there were 7,830 cell phone users in the city.
Let t represent the number of years since 1995. Assuming the number of cell phone users increased exponentially, which expression best models the number of cell phone users in Northridge over time?
According to this model, how many cell phone users did Northridge have in 2000?
Round your answer to the nearest whole number.
cell phone users
To find the expression that models the number of cell phone users in Northridge over time, we can use the formula for exponential growth:
N(t) = N₀ * e^(kt)
Where:
N(t) is the number of cell phone users at time t
N₀ is the starting number of cell phone users (in 1995)
e is Euler's number (approximately 2.71828)
k is the growth rate per year
We can find k by using the formula:
k = ln(N₁/N₀) / (t₁ - t₀)
Where:
N₁ is the number of cell phone users in the following year (1996)
t₁ is the following year (1996)
t₀ is the starting year (1995)
Given N₀ = 5,400, N₁ = 7,830, t₀ = 1995, and t₁ = 1996, we can substitute the values into the formula to find k:
k = ln(7,830/5,400) / (1996 - 1995)
k ≈ ln(1.45) ≈ 0.3716
Now we can substitute N₀ = 5,400 and k = 0.3716 into the formula N(t) = N₀ * e^(kt):
N(t) = 5400 * e^(0.3716t)
To find the number of cell phone users in Northridge in 2000 (t=5), we can substitute t = 5 into the expression:
N(5) = 5400 * e^(0.3716*5)
N(5) ≈ 5400 * e^(1.858)
N(5) ≈ 5400 * 6.3984
N(5) ≈ 34,622.66
Rounding to the nearest whole number, Northridge had approximately 34,623 cell phone users in 2000.