The percentage of research articles in a prominent journal written by researchers in the United States can be modeled by
A(t) = 26 + (36/1 + 0.8(0.8)^−t'),
where t is time in years (t = 0 represents 1983). Numerically estimate
lim as t→+infinity A(t).
I calculated
A(t)= 26 + (36/1 + 0.8(0.8)^−1983)
and got nothing.(0.8(0.8)^−1983)<- this doesn't work)
Also
Interpret the answer. (Does it have anything to do with the answer? The percentage I mean, if there is, how do i determine the answer below?)
A.In the long term, the percentage of research articles in the journal written by researchers in the U.S. approaches 26%.
B.In the long term, the percentage of research articles in the journal written by researchers in the U.S. approaches 36%.
C.In the long term, the number of research articles in the journal written by researchers in the U.S. approaches 26
D.In the long term, the percentage of research articles in the journal written by researchers in the U.S. approaches 0%.
Don't know why its wrong. Help Please. Thank you.
if your a(t) is correct, then your limit should be 62. from what i see, you have 26+36+.64^-t. if that is the case, then you should combine like terms and then you should have 62+.64^-t. Well that .64^-t will be 1/.64^t and when you apply the infinity and its under 1 that resultant will always be 0 leaving you with the answer being 62
To numerically estimate the limit as t approaches infinity for A(t), we need to rewrite the expression in the equation accordingly.
Given A(t) = 26 + (36/(1 + 0.8(0.8)^-t)), the term (0.8)^-t can be simplified using exponential properties. Remember that any base raised to a negative exponent is equivalent to 1 divided by the base raised to the positive exponent. Therefore, (0.8)^-t can be rewritten as 1/(0.8)^t.
Substituting this back into the equation, we have:
A(t) = 26 + (36/(1 + 0.8/(0.8)^t))
Now, we can calculate the value of A(t) as t approaches infinity by plugging in very large values of t and observing the trend. Let's try plugging in some large values:
A(100) ≈ 26 + (36/(1 + 0.8/(0.8)^100))
A(1000) ≈ 26 + (36/(1 + 0.8/(0.8)^1000))
A(10000) ≈ 26 + (36/(1 + 0.8/(0.8)^10000))
As t gets larger and larger, the term (0.8)^t becomes negligible compared to 1, and the denominator of the fraction approaches 1. Thus, we can simplify the expression to:
A(t) ≈ 26 + (36/1) = 26 + 36 = 62
Therefore, the estimated value of A(t) as t approaches infinity is approximately 62.
Regarding the interpretation of the answer, it corresponds to the percentage of research articles in the journal written by researchers in the United States. The value of 62 implies that, in the long term, around 62% of the research articles in the prominent journal are expected to be written by researchers in the United States. So, the correct answer would be:
B. In the long term, the percentage of research articles in the journal written by researchers in the U.S. approaches 36%.