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Answer :
To solve this problem, we need to find out how many years it will take for the deer population to reach 3000, using the growth model formula:
[tex]\[ P = 578 \times e^{0.045t} \][/tex]
where [tex]\( P \)[/tex] represents the population, and [tex]\( t \)[/tex] is the time in years.
1. Set up the equation with the target population:
[tex]\( 3000 = 578 \times e^{0.045t} \)[/tex]
2. Isolate the exponential part:
Divide both sides by 578 to simplify the equation:
[tex]\( e^{0.045t} = \frac{3000}{578} \)[/tex]
3. Convert to a logarithmic equation:
Use the natural logarithm to solve for [tex]\( t \)[/tex]:
[tex]\( \ln(e^{0.045t}) = \ln\left(\frac{3000}{578}\right) \)[/tex]
This simplifies to:
[tex]\( 0.045t = \ln\left(\frac{3000}{578}\right) \)[/tex]
4. Solve for [tex]\( t \)[/tex]:
Divide by 0.045 to find [tex]\( t \)[/tex]:
[tex]\( t = \frac{\ln\left(\frac{3000}{578}\right)}{0.045} \)[/tex]
5. Calculate [tex]\( t \)[/tex]:
Performing the above calculation will give you:
[tex]\( t \approx 36.6 \)[/tex]
This means it will take approximately 36.6 years for the deer population to reach 3000. The correct answer is option (A) 36.6 years.
[tex]\[ P = 578 \times e^{0.045t} \][/tex]
where [tex]\( P \)[/tex] represents the population, and [tex]\( t \)[/tex] is the time in years.
1. Set up the equation with the target population:
[tex]\( 3000 = 578 \times e^{0.045t} \)[/tex]
2. Isolate the exponential part:
Divide both sides by 578 to simplify the equation:
[tex]\( e^{0.045t} = \frac{3000}{578} \)[/tex]
3. Convert to a logarithmic equation:
Use the natural logarithm to solve for [tex]\( t \)[/tex]:
[tex]\( \ln(e^{0.045t}) = \ln\left(\frac{3000}{578}\right) \)[/tex]
This simplifies to:
[tex]\( 0.045t = \ln\left(\frac{3000}{578}\right) \)[/tex]
4. Solve for [tex]\( t \)[/tex]:
Divide by 0.045 to find [tex]\( t \)[/tex]:
[tex]\( t = \frac{\ln\left(\frac{3000}{578}\right)}{0.045} \)[/tex]
5. Calculate [tex]\( t \)[/tex]:
Performing the above calculation will give you:
[tex]\( t \approx 36.6 \)[/tex]
This means it will take approximately 36.6 years for the deer population to reach 3000. The correct answer is option (A) 36.6 years.
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