Discover a wealth of information and get your questions answered on IDNLearn.com. Discover thorough and trustworthy answers from our community of knowledgeable professionals, tailored to meet your specific needs.
Sagot :
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]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for trusting IDNLearn.com. We’re dedicated to providing accurate answers, so visit us again for more solutions.