I don't know how to go about this question. The number of elephants in a park is estimated to be
p(t)=7500/(1+749e^-0.15t)
Where t is the time in years and t=0 corresponds to the year 1930. Find the inverse p(t) and the interpretation of that.
http://www.jiskha.com/display.cgi?id=1453991598
inverse would be time for a specific population
1+749 e^-.15t = 7500/p
749 e^-0.15 t = (7500-p)/p
e^-.015 t = (7500-p)/(749 p)
e^.015t = 749 p/(7500-p)
.015 t = ln [ 749 p] - ln [7500-p)]
t = 66.7 [ln 749 p - ln(7500-p)]
we had 10 when t was 0
so check
t = 66.7[ ln 7490 - ln 7490 ]
so yes, at least it checks when t = 0
To find the inverse of the function p(t), we need to switch the roles of t and p(t). Let's denote the inverse of p(t) as t(p).
To do this, we start with the function: p(t) = 7500 / (1 + 749e^(-0.15t))
1. Swap the variables:
t(p) = 7500 / (1 + 749e^(-0.15t(p)))
2. Solve for t(p):
Multiply both sides by (1 + 749e^(-0.15t(p))):
t(p) * (1 + 749e^(-0.15t(p))) = 7500
3. Simplify the equation:
t(p) + 749e^(-0.15t(p)) * t(p) = 7500
t(p) + 749t(p) * e^(-0.15t(p)) = 7500
4. Rearrange the equation:
t(p) * (1 + 749e^(-0.15t(p))) = 7500
Now, unfortunately, there isn't a simple algebraic expression for t(p) in terms of elementary functions. The inverse function t(p) cannot be expressed in a straightforward manner. However, we can use numerical methods or graphing calculators to find an approximate solution for t(p) given a specific value of p.
Interpretation:
The inverse function t(p) represents the number of years that have passed since 1930 (t=0) when the population of elephants in the park is estimated to be p. In other words, if you have a specific number of elephants in mind, the inverse function will give you an estimation of when that population was reached based on the given growth model.