The following logistic function describes the percent of the population of a city that will purchase the local newspaper.
P(t)=(65)/(1+34e^(-0.223 t))
"t" is the number of days since the newspaper is first launched.
When will the newspaper reach 60% of the population
Solve 60 = (65)/(1+34e^(-0.223 t))
(1+34e^(-0.223 t)) = 65/60 = 1.0833
34 e^(-0.223 t) = 0.0833
e^(-0.223 t) = 2.25*10^-3
-.223 t = -6.01
t = 27 days
To find when the newspaper will reach 60% of the population, we need to solve for t in the equation P(t) = 0.6, where P(t) is the percent of the population purchasing the newspaper at time t.
The logistic function given is P(t) = 65 / (1 + 34e^(-0.223t)). Set P(t) equal to 0.6:
0.6 = 65 / (1 + 34e^(-0.223t))
To solve for t, we will need to isolate the exponential term on one side of the equation. Start by multiplying both sides of the equation by (1 + 34e^(-0.223t)):
0.6(1 + 34e^(-0.223t)) = 65
Next, distribute 0.6 to the terms inside the parentheses:
0.6 + 20.4e^(-0.223t) = 65
Now, isolate the exponential term by subtracting 0.6 from both sides:
20.4e^(-0.223t) = 65 - 0.6
20.4e^(-0.223t) = 64.4
Divide both sides by 20.4:
e^(-0.223t) = 64.4 / 20.4
e^(-0.223t) = 3.16078
To isolate the base of the natural logarithm, take the natural logarithm of both sides:
ln(e^(-0.223t)) = ln(3.16078)
The natural logarithm and the base of the exponent cancel each other out:
-0.223t = ln(3.16078)
Now, divide both sides by -0.223 to solve for t:
t = ln(3.16078) / -0.223
Using a calculator, find the value of ln(3.16078) and divide it by -0.223. The result will be the value of t when the newspaper reaches 60% of the population.