Answer :
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]
[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]
