I do not know how to solve for y to get the inverse of 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 1903. Find the inverse of P(t). What is the interpretation of this inverse?
well, I don't know what "7500 1" means, but I'll just call it 7500 and you can fix it as needed.
For the inverse, swap variables and solve for y:
t = 7500 + 749e^(-0.15y)
t-7500 = 749e^(-0.15y)
e^(-0.15y) = (t-7500)/749
-0.15y = log[(t-7500)/749]
y = -log[(t-7500)/749]/0.15
Clearly, if P is the number of elephants t years after 1903, t is the number of years to arrive at P elephants.
To find the inverse of the function P(t) = 7500(1 + 749e^(-0.15t)), we need to switch the roles of t and P(t) and solve for t.
Step 1: Switch the roles of t and P(t)
Let y = 7500(1 + 749e^(-0.15t))
Step 2: Solve for t in terms of y
Now we need to isolate t on one side of the equation.
Divide both sides by 7500:
y/7500 = 1 + 749e^(-0.15t)
Subtract 1 from both sides:
(y/7500) - 1 = 749e^(-0.15t)
Divide both sides by 749:
((y/7500) - 1) / 749 = e^(-0.15t)
Take the natural logarithm (ln) of both sides to eliminate the exponential function:
ln(((y/7500) - 1) / 749) = -0.15t
Divide both sides by -0.15 to solve for t:
t = -ln(((y/7500) - 1) / 749) / 0.15
So the inverse function of P(t), denoted as P^(-1)(y), is:
P^(-1)(y) = -ln(((y/7500) - 1) / 749) / 0.15
The interpretation of this inverse function is that given the estimated number of elephants in a park (y), the inverse function allows us to determine the corresponding time (t) in years when there were approximately that many elephants. It provides a way to find the year, relative to 1903, when a specific number of elephants was estimated to be in the park.