There were 8.65 million licensed drivers in Pennsylvania in 2009 and 8.43 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 ->
b) exponential ->
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a) Linear Growth:
To find a formula for linear growth, we can use the slope-intercept form of a linear equation: y = mx + b, where y represents the dependent variable, x represents the independent variable, m represents the slope, and b represents the y-intercept.
In this case, we can assume that the number of licensed drivers in Pennsylvania grows linearly over time. Let's calculate the slope (m) first:
m = (change in y) / (change in x)
m = (8.65 million - 8.43 million) / (2009 - 2004)
m = 220,000 / 5
m = 44,000
Now, we can find the y-intercept (b) by plugging in the values of one of the given years:
8.43 million = 44,000(2004) + b
8.43 million = 88,176,000 + b
b = 8.43 million - 88,176,000
b = -79,746,000
Therefore, the formula for the number (N) of licensed drivers in the US as a function of t (the number of years since 2004) assuming linear growth is:
N = 44,000t - 79,746,000
b) Exponential Growth:
To find a formula for exponential growth, we can use the formula: N = N₀ e^(kt), where N represents the number of licensed drivers in the US, N₀ represents the initial number of licensed drivers, k represents the growth constant, and t represents the number of years since 2004.
Let's calculate the growth constant (k) first:
8.43 million = N₀ e^(k * 0) (using the year 2004 as the initial year)
8.65 million = N₀ e^(k * 5) (using the year 2009)
Dividing the second equation by the first equation, we get:
(8.65 million) / (8.43 million) = e^(k * 5) / e^(k * 0)
1.025835 = e^(5k)
Taking the natural logarithm (ln) of both sides, we get:
ln(1.025835) = ln(e^(5k))
ln(1.025835) = 5k
Solving for k, we find:
k = ln(1.025835) / 5
k ≈ 0.00515
Therefore, the formula for the number (N) of licensed drivers in the US as a function of t (the number of years since 2004) assuming exponential growth is:
N = N₀ e^(0.00515t)
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 can use the formula for the equation of a line:
N(t) = mt + b
where m is the rate of change (slope) and b is the initial value (y-intercept).
First, we need to find the slope (m) using the data given. The change in the number of licensed drivers from 2004 to 2009 is:
Change in N = N(2009) - N(2004)
= 8.65 million - 8.43 million
= 0.22 million
The change in time (t) is 5 years (2009 - 2004).
Now, we can calculate the slope:
m = Change in N / Change in t
= 0.22 million / 5 years
= 0.044 million/year
Next, we need to determine the initial value (b). We can use either data point given. Let's use the data from 2004:
N(0) = 8.43 million
Finally, we can write the formula for the number of licensed drivers in the US as a function of time (t) with linear growth:
N(t) = 0.044t + 8.43 million
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 can use the equation for exponential growth:
N(t) = N(0) * e^(kt)
Where N(0) is the initial value (number of licensed drivers in 2004), e is Euler's number (approximately 2.71828), k is the constant growth rate, and t is the number of years since 2004.
To find the constant growth rate (k), we can use the given data. The change in the number of licensed drivers from 2004 to 2009 is:
Change in N = N(2009) - N(2004)
= 8.65 million - 8.43 million
= 0.22 million
The change in time (t) is 5 years (2009 - 2004).
Now, we can calculate the constant growth rate (k):
k = ln(N(2009)/N(2004)) / t
= ln(8.65 million/8.43 million) / 5 years
≈ 0.0416 per year
Finally, we can write the formula for the number of licensed drivers in the US as a function of time (t) with exponential growth:
N(t) ≈ 8.43 million * e^(0.0416t)
a) To find a formula for the number of licensed drivers in the US as a function of time, assuming linear growth, we can use the point-slope form of a linear equation:
N(t) = m * t + b
Where N(t) represents the number of licensed drivers in the US at time t, m represents the rate of change (slope), and b represents the initial value (the number of licensed drivers in 2004).
To find the slope (m), we can use the formula:
m = (N2 - N1) / (t2 - t1)
Given that there were 8.65 million licensed drivers in Pennsylvania in 2009 (t2 = 2009 - 2004 = 5) and 8.43 million in 2004 (t1 = 0), we can substitute these values into the formula:
m = (8.65 - 8.43) / (5 - 0) = 0.04
So, the slope (m) is 0.04.
Next, we can substitute the initial value (b) using the number of licensed drivers in 2004:
N(0) = 0.04 * 0 + b
8.43 = b
Therefore, the equation for the number of licensed drivers in the US as a function of time, assuming linear growth, is:
N(t) = 0.04t + 8.43
b) To find a formula for the number of licensed drivers in the US as a function of time, assuming exponential growth, we can use an exponential function of the form:
N(t) = N0 * e^(kt)
Where N(t) represents the number of licensed drivers in the US at time t, N0 represents the initial value (the number of licensed drivers in 2004), e is the base of the natural logarithm (approximately 2.71828), k represents the growth rate constant.
To find the growth rate constant (k), we can use the formula:
k = ln(N(t) / N0) / t
Given that there were 8.43 million licensed drivers in 2004 (N0) and assuming the number of licensed drivers in the US (N(t)) increases exponentially over time from 2004 to 2009 (t = 5), we can substitute these values into the formula:
k = ln(8.65 / 8.43) / 5 ≈ 0.0124
So, the growth rate constant (k) is approximately 0.0124.
Therefore, the equation for the number of licensed drivers in the US as a function of time, assuming exponential growth, is:
N(t) = 8.43 * e^(0.0124t)
What, you forget your algebra?
(a) N = 8.43 + (8.65-8.43)/(2009-2004)t
(b) N = 8.43 * (8.65/8.43)^(t/(2009-2004))