To find the exponential growth rate [tex]\( k \)[/tex], we can use the formula for exponential growth:
[tex]\[ V(t) = V_0 \cdot e^{kt} \][/tex]
where:
- [tex]\( V(t) \)[/tex] is the value at time [tex]\( t \)[/tex],
- [tex]\( V_0 \)[/tex] is the initial value,
- [tex]\( k \)[/tex] is the growth rate,
- [tex]\( t \)[/tex] is the time.
Given:
- The initial value in 1980, [tex]\( V_0 = 212 \)[/tex] dollars,
- The value in 1990, [tex]\( V(t) = 418 \)[/tex] dollars,
- The time period [tex]\( t = 1990 - 1980 = 10 \)[/tex] years.
We need to find the value of [tex]\( k \)[/tex]. Start by substituting the known values into the exponential growth formula:
[tex]\[ 418 = 212 \cdot e^{k \cdot 10} \][/tex]
To isolate [tex]\( e^{k \cdot 10} \)[/tex], divide both sides by 212:
[tex]\[ \frac{418}{212} = e^{k \cdot 10} \][/tex]
Simplify the fraction:
[tex]\[ 1.9717 \approx e^{10k} \][/tex]
Next, take the natural logarithm (ln) of both sides to solve for [tex]\( k \)[/tex]:
[tex]\[ \ln(1.9717) = \ln(e^{10k}) \][/tex]
Using the property of logarithms that [tex]\( \ln(e^x) = x \)[/tex]:
[tex]\[ \ln(1.9717) = 10k \][/tex]
Solve for [tex]\( k \)[/tex] by dividing both sides by 10:
[tex]\[ k = \frac{\ln(1.9717)}{10} \][/tex]
Using the numerical value of [tex]\( \ln(1.9717) \approx 0.678 \)[/tex]:
[tex]\[ k \approx \frac{0.678}{10} \][/tex]
[tex]\[ k \approx 0.0678 \][/tex]
Rounding to the nearest thousandth:
[tex]\[ k \approx 0.068 \][/tex]
Thus, the exponential growth rate [tex]\( k \)[/tex] is approximately [tex]\( 0.068 \)[/tex] (rounded to the nearest thousandth).
