To determine how many years it will take for the farmland's market value to reach [tex]$125,000$[/tex] dollars, we start with the given exponential growth function:
[tex]\[ p(t) = 78,125 \cdot e^{0.025t} \][/tex]
We need to find [tex]\( t \)[/tex] when the market value [tex]\( p(t) \)[/tex] is [tex]$125,000.
First, set the function equal to the target value:
\[ 125,000 = 78,125 \cdot e^{0.025t} \]
To isolate the exponential term, divide both sides by 78,125:
\[ \frac{125,000}{78,125} = e^{0.025t} \]
Simplify the left-hand side:
\[ \frac{125,000}{78,125} = 1.6 \]
So, we have:
\[ 1.6 = e^{0.025t} \]
Next, take the natural logarithm of both sides to solve for \( t \):
\[ \ln(1.6) = \ln(e^{0.025t}) \]
Since \( \ln(e^x) = x \), this simplifies to:
\[ \ln(1.6) = 0.025t \]
Finally, solve for \( t \) by dividing both sides by 0.025:
\[ t = \frac{\ln(1.6)}{0.025} \]
Using the calculated result:
\[ t \approx 18.8 \]
Therefore, the number of years it will take for the farmland's market value to reach $[/tex]125,000 is approximately:
[tex]\[ t \approx 18.8 \, \text{years} \][/tex]
