SOLUTION: R(t) = 1150/0.5+22.5(2.2)^-0.065t the exponent is -0.065t t = is the number of weeks after the research team first introduced the rabbits into the forest.

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: R(t) = 1150/0.5+22.5(2.2)^-0.065t the exponent is -0.065t t = is the number of weeks after the research team first introduced the rabbits into the forest.       Log On


   

Question 426300: R(t) = 1150/0.5+22.5(2.2)^-0.065t
the exponent is -0.065t
t = is the number of weeks after the research team first introduced the rabbits into the forest.
I figured out the below answers and they are right, I cannot figure out this, the rabbit population approaches ________ as time goes on?

the research team brought 50 rabbits to the forest (t=0)
how many rabbits can be expected after 10weeks = 82 rabbits (t=10)
how many rabbits can be expected after the first year = 557 rabbits (t=52)
how many rabbits can be expected after 4 years = 2298 rabbits (t=208)
how many rabbits can be expected after 5 years = 2300 rabbits (t=260)

Answer by jsmallt9(3759) About Me  (Show Source):
You can put this solution on YOUR website!
I assume the function is:
R%28t%29+=+1150%2F%280.5%2B22.5%282.2%29%5E%28-0.065t%29%29

To answer the question about the rabbit population approaching some number "as time goes on" it means, in mathematical terms, as t approaches infinity.

To answer this question we use some logic and some knowledge about how exponents and fractions work:
  • As t approaches infinity (i.e a very, very large number), the exponent, -0.065t, will be become a very, very large negative number.
  • Since negative exponents mean reciprocals, 2.2 to a very large negative power is a fraction: 1 over 2.2 to a very, very large positive exponent.
  • As t gets larger and larger, the denominator of this fraction gets larger and larger.
  • As the denominator of a fraction gets larger and larger (without a change in the numerator), the fraction gets smaller and smaller. In fact it gets closer and closer to zero.
  • So as t gets larger and larger, 2.2%5E%28-0.065t%29 gets closer and closer to zero.
  • As 2.2%5E%28-0.065t%29 gets closer to zero, 22.5%2A2.2%5E%28-0.065t%29 also gets closer and closer to zero.
  • As 22.5%2A2.2%5E%28-0.065t%29 gets closer and closer to zero, the denominator, 0.5+%2B+22.5%2A2.2%5E%28-0.065t%29, gets closer and closer to 0.5.
  • As the denominator gets closer and closer to 0.5, R(t) gets closer and closer to 1150%2F0.5
  • And since 1150%2F0.5+=+2300, the population of rabbits will approach 2300 "as time goes by".

P.S. R(t) only approaches 2300 for large values of t. It will never actually be equal to 2300. When you got an answer of 2300 for the population after 5 years you must have rounded off your answer to get 2300 (or you made an error.)