To find the average rate of change of the investment's value between 2014 and 2016, we need to follow these steps:
1. Define the function for the value of the investment:
The value [tex]\( V \)[/tex] of the investment [tex]$ t $[/tex] years after 2012 is given by:
[tex]\[
V(t) = 1000 \cdot \left(2^{1/6}\right)^t
\][/tex]
2. Determine the values for the specific years:
We need to find the value of the investment in the years 2014 and 2016. First, calculate how many years after 2012 these years are.
- For 2014: [tex]\( t_{2014} = 2014 - 2012 = 2 \)[/tex]
- For 2016: [tex]\( t_{2016} = 2016 - 2012 = 4 \)[/tex]
3. Calculate the value of the investment for 2014 and 2016:
Since [tex]\( t_{2014} = 2 \)[/tex]:
[tex]\[
V(2014) = 1000 \cdot \left(2^{1/6}\right)^2 = 1000 \times 2^{1/3} \approx 1259.92
\][/tex]
Since [tex]\( t_{2016} = 4 \)[/tex]:
[tex]\[
V(2016) = 1000 \cdot \left(2^{1/6}\right)^4 = 1000 \times 2^{2/3} \approx 1587.40
\][/tex]
4. Calculate the average rate of change:
The average rate of change of the investment between 2014 and 2016 is given by the formula:
[tex]\[
\text{Average Rate of Change} = \frac{V(t_{2016}) - V(t_{2014})}{t_{2016} - t_{2014}}
\][/tex]
Plug in the values we calculated:
[tex]\[
\text{Average Rate of Change} = \frac{1587.40 - 1259.92}{4 - 2} = \frac{327.48}{2} \approx 163.74
\][/tex]
5. Round the final result to two decimal places:
The average rate of change of the investment's value between 2014 and 2016 is approximately [tex]$ 163.74 $[/tex] dollars per year.
Thus, the average rate of change of the investment's value between 2014 and 2016 is approximately:
[tex]\[
\boxed{163.74 \text{ dollars/year}}
\][/tex]
