Solve the problem.
The growth in the population of a certain rodent at a dump site fits the exponential function A(t)= 164e0.032t, where t is the number of years since 1975. Estimate the
population in the year 2000.
Hint: figure out how many years would have passed and substitute it for t.
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To estimate the population in the year 2000, we need to find the value of the exponential function A(t) when t is equal to 2000 - 1975 = 25.
We'll substitute t = 25 into the function A(t) = 164e^0.032t:
A(25) = 164e^(0.032*25)
We calculate this using a calculator or a math software:
A(25) ≈ 164e^(0.8)
Using the approximate value of e ≈ 2.71828:
A(25) ≈ 164 * 2.71828^0.8
Simplifying this expression:
A(25) ≈ 164 * 2.2255
A(25) ≈ 364.493
Therefore, the estimated population in the year 2000 is approximately 364.493 rodents.
To estimate the population in the year 2000, we need to substitute t=2000-1975=25 into the exponential function A(t)=164e^(0.032t). Here's how you can calculate it:
Step 1: Substitute the value of t into the exponential function: A(25) = 164e^(0.032 * 25).
Step 2: Calculate the value of the exponent: 0.032 * 25 = 0.8.
Step 3: Calculate e^(0.8). The value of e is approximately 2.71828, so e^(0.8) is approximately 2.22554.
Step 4: Multiply 164 by the result of step 3: 164 * 2.22554 = 365.07336.
Therefore, the estimate of the population in the year 2000 is approximately 365.07336 rodents.
Keep in mind that this is an estimate based on the given exponential growth model. The actual population in the year 2000 may vary.