Find the logistic function that satisfies the given conditions.
15) Initial value = 35, limit to growth = 140, passing through (1, 56)
Thank you!
Given your parameters, you have
y(t) = (35*140)/(35+(140-35)e^(-rt)) = 4900/(35+105e^(-rt))
So, using y(1)=56,
4900/(35+105e^(-4r)) = 56
r = 0.1733
That makes y(t) = 4900/(35+105e^(-0.1733t))
Why did the logistic function bring a pen and paper to this problem?
Because it wanted to "plot" the solution!
Now let's find the logistic function that satisfies the given conditions. The general form of a logistic function is:
f(x) = L / (1 + e^(-k(x - x0)))
Where:
L = Limit to growth
k = Growth rate constant
x0 = x-coordinate of the midpoint of the sigmoid curve
Given that the initial value is 35 and the limit to growth is 140, we have:
f(0) = 35
f(∞) = 140
To find the growth rate constant, we need to use the passing through point (1, 56). Let's plug in these values:
56 = L / (1 + e^(-k(1 - x0)))
Now we can solve for x0:
1 + e^(-k(1 - x0)) = L / 56
Taking the natural logarithm (ln) of both sides:
ln(1 + e^(-k(1 - x0))) = ln(L / 56)
-k(1 - x0) = ln(L / 56) - 1
k(1 - x0) = 1 - ln(L / 56)
Finally, we can substitute these values into the logistic function:
f(x) = 140 / (1 + e^(-(1 - x0)(1 - ln(140 / 56))))
To find the logistic function that satisfies the given conditions, we can use the logistic growth formula:
P(t) = L / (1 + A * e^(-k*t))
where P(t) represents the population at time t, L is the limit to growth or carrying capacity, A is the initial value minus the limit to growth (A = P(0) - L), k is the growth rate constant, and e is the base of the natural logarithm.
From the given conditions, we have:
Initial value (P(0)) = 35
Limit to growth (L) = 140
Passing through (1, 56)
Let's substitute these values into the formula:
56 = 140 / (1 + A * e^(-k*1))
To simplify further, let's substitute A = P(0) - L:
56 = 140 / (1 + (35 - 140) * e^(-k*1))
Simplifying, we have:
56 = 140 / (1 - 105 * e^(-k))
After isolating the exponential term, we get:
(1 - 105 * e^(-k)) / 140 = 1 / 56
Simplifying further, we have:
1 - 105 * e^(-k) = 140 / 56
1 - 105 * e^(-k) = 2.5
Rearranging the equation:
105 * e^(-k) = -1.5
Taking the natural logarithm of both sides:
ln(105 * e^(-k)) = ln(-1.5)
Using logarithm properties:
ln(105) + ln(e^(-k)) = ln(-1.5)
Simplifying:
ln(105) - k = ln(-1.5)
Now, we can solve for k:
k = -ln(-1.5) + ln(105)
Using a calculator:
k ≈ 0.1986
Now that we have k, we can substitute it back into the equation to find A:
1 - 105 * e^(-k) = 2.5
105 * e^(-k) = -1.5
e^(-0.1986) ≈ -1.5 / 105
e^(-0.1986) ≈ -0.0143
Taking the natural logarithm of both sides:
-0.1986 = ln(-0.0143)
This is not possible because the natural logarithm is undefined for negative values. Thus, there might be a mistake in the given conditions, or there might not be a logistic function that satisfies these conditions.
To find the logistic function that satisfies the given conditions, we can use the formula for the logistic function:
f(t) = L / (1 + C * e^(-k(t-t0)))
Where:
f(t) is the value of the function at time t
L is the limit to growth
C is the initial value minus the limit to growth (C = Initial value - L)
k is a constant that determines the rate of growth
t0 is the time at which the initial value occurs
Given conditions:
Initial value (C) = 35
Limit to growth (L) = 140
Point on the graph: (1, 56)
To find the value of k, we can substitute the values of t, f(t) and C into the equation and solve for k:
56 = 140 / (1 + C * e^(-k(1-t0)))
Substituting the values of t=1, f(t)=56, C=35, and L=140:
56 = 140 / (1 + (35-140) * e^(-k(1-t0)))
Simplifying the equation:
56 = 140 / (1 - 105 * e^(-k(1-t0)))
Rearranging the equation:
1 - 105 * e^(-k(1-t0)) = 140 / 56
1 - 105 * e^(-k(1-t0)) = 2.5
Now, we can solve for e^(-k(1-t0)):
105 * e^(-k(1-t0)) = -1.5
e^(-k(1-t0)) = -1.5 / 105 = -0.0142857 (approximately)
Taking the natural logarithm of both sides of the equation:
-k(1-t0) = ln(-0.0142857)
k(1-t0) = -ln(-0.0142857)
Now we can calculate the value of k:
k = -ln(-0.0142857) / (1-t0)
Substituting the value of t0=1:
k = -ln(-0.0142857) / (1-1) = undefined (since we get a division by zero, we conclude that there is no solution).
In this case, it seems that there is no logistic function that satisfies the given conditions.