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The table shows the number of rabbits r in a particular forest t years after a forest fire. Write and use an exponential model to find how many years it will take for the rabbit population to surpass 20,000. Please show your work!​

The table shows the number of rabbits r in a particular forest t years after a forest-example-1
User Jvdub
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1 Answer

2 votes

Answer:

t = 6.29 years

Explanation:

The general formula of an exponential function is


y=ab^x, where a is the initial value (i.e., value of y when x = 0), b is the base, and x is the exponent.

We can express rabbit population as a function of time in years, which means in our formula, r is like y and t is like x:


r=ab^t

Because at time t = 0, the rabbit population is 20, we know that our a value for the equation is 20.

We can find b by simply plugging in 20 for a and any point for r and t like (1, 60):


60=20b^1\\3=b

Thus, the equation we will need to use to find the rabbit population is


r < 20(3)^t

It's helpful to use an inequality where the equation is greater than r since we want to know how many years it will take for the population to exceed 20000:


20000 < 20(3)^t\\\\1000 < 3^t\\\\log(1000) < log(3)^t\\\\log(1000) < t*log(3)\\\\log(1000)/log(3) < t\\\\6.287709823 < t\\6.29 < t

We can check our answer by plugging in 6.29 for t and check whether the answer we get is greater than 20000:


20000 < 20(3)^6^.^2^9\\20000 < 20*1002.519185\\20000 < 20050.38369\\20000 < 20050.38

User BZink
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