There were 15.37 million licensed drivers in Texas in 2009 and 14.54 million in 2004. Find a formula for the number, N, of licensed drivers in the US as a function of t, the number of years since 2004, assuming growth is
(a) Linear
N(t)= million drivers
(b) Exponential
N(t)= million drivers
If we consider 2004 to be represented by 0,
then 2009 would correspond with 5
and we have two ordered pairs (0,14.54) and (5,15.37)
slope = (15.37-14.54)/(5-0) = .83
N = .83t + c
at (0,14.54)
14.54 = 0 + c , c = 14.54
N = .83t + 14.54 , where t is in years since 2004 and N is in millions of drivers
You don't specify what type of exponential, (what base), so I will use base e
N = a e^kt, were a is the initial number and t is the time since 2004
N = 14.54 e^kt
when t = 5
15.37=14.54 e^(5k)
1.057... = e^5k
using ln
5k = ln 1.057..
k = .0111028.. or appr .0111
N(t) = 14.54 e^(.0111 t)
(a) Linear growth:
To find a formula for the number of licensed drivers in the US as a function of time (t) assuming linear growth, we'll use the data from 2004 and 2009 as reference points to determine the slope of the line.
Let's denote the number of licensed drivers in a given year as N(t), and the number of years since 2004 as t. Using the data from 2004 and 2009:
N(t=0) = 14.54 million drivers in 2004
N(t=5) = 15.37 million drivers in 2009
Since we have two points, we can determine the slope of the linear function:
slope = (N(t=5) - N(t=0)) / (t=5 - t=0)
slope = (15.37 - 14.54) / (5 - 0)
slope = 0.167 million drivers per year
Now that we know the slope, we can use the point-slope form of a linear equation to find the formula for N(t):
N(t) = N(t=0) + slope * t
Substituting the known values:
N(t) = 14.54 + 0.167t million drivers
Therefore, the formula for the number of licensed drivers in the US as a function of time (t) assuming linear growth is:
N(t) = 14.54 + 0.167t million drivers
(b) Exponential growth:
To find a formula for the number of licensed drivers in the US as a function of time (t) assuming exponential growth, we'll use the data from 2004 and 2009 as reference points to determine the growth rate.
Let's denote the number of licensed drivers in a given year as N(t), and the number of years since 2004 as t. Using the data from 2004 and 2009:
N(t=0) = 14.54 million drivers in 2004
N(t=5) = 15.37 million drivers in 2009
Since we have two points, we can determine the growth rate using the exponential growth formula:
Growth rate = (N(t=5) / N(t=0)) ^ (1 / t)
Growth rate = (15.37 / 14.54) ^ (1 / 5)
Growth rate = 1.032
Now that we know the growth rate, we can use the exponential growth formula to find the formula for N(t):
N(t) = N(t=0) * (Growth rate) ^ t
Substituting the known values:
N(t) = 14.54 * (1.032) ^ t million drivers
Therefore, the formula for the number of licensed drivers in the US as a function of time (t) assuming exponential growth is:
N(t) = 14.54 * (1.032) ^ t million drivers
(a) For linear growth, we can assume that the number of licensed drivers in Texas follows a linear trend over time. To find a formula for the number of licensed drivers in the US as a function of years since 2004 (t), we need to calculate the average annual increase in the number of licensed drivers in Texas.
Let's find the average annual increase in the number of licensed drivers:
Change in the number of licensed drivers = 15.37 million - 14.54 million = 0.83 million drivers
Change in years = 2009 - 2004 = 5 years
Average annual increase = Change in the number of licensed drivers / Change in years
= 0.83 million drivers / 5 years
= 0.166 million drivers per year
Now, let's find the formula for the number of licensed drivers in the US as a function of years since 2004:
N(t) = 14.54 million + (0.166 million drivers per year) * t
= 14.54 + 0.166t million drivers
Therefore, the formula for the number of licensed drivers in the US, assuming linear growth, is N(t) = 14.54 + 0.166t million drivers.
(b) For exponential growth, we can assume that the number of licensed drivers in Texas follows an exponential trend over time. To find a formula for the number of licensed drivers in the US as a function of years since 2004 (t), we need to calculate the growth rate.
Let's calculate the growth rate:
Growth rate = (Number of licensed drivers in 2009 / Number of licensed drivers in 2004) ^ (1 / Change in years) - 1
= (15.37 million / 14.54 million) ^ (1 / 5) - 1
≈ 1.0572 - 1
≈ 0.0572
Now, let's find the formula for the number of licensed drivers in the US as a function of years since 2004:
N(t) = Number of licensed drivers in 2004 * (1 + Growth rate) ^ t
= 14.54 million * (1 + 0.0572) ^ t
≈ 14.54 * 1.0572^t million drivers
Therefore, the formula for the number of licensed drivers in the US, assuming exponential growth, is N(t) ≈ 14.54 * 1.0572^t million drivers.
(a) Linear growth assumes that the number of licensed drivers increases by a fixed amount each year. To find the formula for the number, N, of licensed drivers in the US as a function of t, the number of years since 2004, we need to determine the growth rate.
First, we calculate the change in the number of licensed drivers from 2004 to 2009:
Change in drivers = Number of drivers in 2009 - Number of drivers in 2004
= 15.37 million - 14.54 million
= 0.83 million
Next, we need to determine the annual growth rate:
Annual growth rate = Change in drivers / Number of years
= 0.83 million / 5 years
= 0.166 million per year
Finally, we can write the formula for linear growth:
N(t) = Number of drivers in 2004 + Annual growth rate * t
N(t) = 14.54 million + 0.166 million * t
Therefore, the formula for the number, N, of licensed drivers in the US as a function of t, assuming linear growth, is:
N(t) = 14.54 million + 0.166 million * t
(b) Exponential growth assumes that the number of licensed drivers increases by a constant percentage each year. To find the formula for the number, N, of licensed drivers in the US as a function of t, the number of years since 2004, we need to determine the annual growth rate.
First, we calculate the growth rate from 2004 to 2009:
Growth rate = (Number of drivers in 2009 / Number of drivers in 2004)^(1 / Number of years) - 1
= (15.37 million / 14.54 million)^(1 / 5) - 1
≈ 0.031 (rounded to 3 decimal places)
Next, we can write the formula for exponential growth:
N(t) = Number of drivers in 2004 * (1 + Growth rate)^t
N(t) = 14.54 million * (1 + 0.031)^t
Therefore, the formula for the number, N, of licensed drivers in the US as a function of t, assuming exponential growth, is:
N(t) = 14.54 million * (1 + 0.031)^t
N(t)=14.54+slope*Y
Y = year-2009
slope= (15.37-14.54)/5
b. N(t)=14.54 e^(kY)
y= year-2009
k= 1/5 * ln(15.37/14.54)