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?
The exponential growth formula is given by:
N(t) = N₀ * e^(kt)
Where:
N(t) is the number of cell phone users at time t
N₀ is the initial number of cell phone users
e is the mathematical constant (approximately equal to 2.71828)
k is the constant of proportionality
t is the number of years since 1995
To find the exponential formula that best models the number of cell phone users in Northridge over time, we need to find the values of N₀ and k.
Given that in 1995 there were 5,400 cell phone users:
N₀ = 5,400
And in 1996 there were 7,830 cell phone users:
N(1) = 7,830
Substituting these values into the equation, we have:
7,830 = 5,400 * e^(k * 1)
Dividing both sides by 5,400, we get:
1.448 = e^k
To solve for k, we take the natural logarithm (ln) of both sides:
ln(1.448) = ln(e^k)
ln(1.448) = k
Thus, the expression that best models the number of cell phone users in Northridge over time is:
N(t) = 5,400 * e^(ln(1.448)t)