Ok, with this one, I'm pretty close. I think I just need another set of eyes to see where I made my mistake.
Here's the problem:
A model for the population P(t) in a suburb of a large city is given by the initial value problem dP/dt = P(10^-1 - 10^-7 P), P(0) = 5000, where t is measured in months. What is the limiting value of the population? At what time will the pop be equal to 1/2 of this limiting value?
//What I know:
a = 10^-1
b = 10^-7
P(0) = 5,000 //initial value
//Answer 1
Limiting value = a/b = 10^-1 / 10^-7 = 1,000,000
1/2 the limiting value is 500,000.
//Plug into the logistic equation
500,000 = 10^-1 * 5,000 / (10^-7 + (10^-1 - 10^-7 * 5,000)e^-t10^-1 )
500,000 = 500 / (0.0005 + 0.0995e^-t/10 )
0.005 + 0.0995e^-t/10 = 0.001
9,9885e^-t/10 = 0.0005
e^-t/10 = 0.0050251
-t/10 = ln(0.0050251)
t = 52.93 months //the answer
I get 52.93 months, but the book tells me the answer is 5.29 months. Somehow, my decimal point is wrong. Please tell me where I made my mistake.
Here's the problem:
A model for the population P(t) in a suburb of a large city is given by the initial value problem dP/dt = P(10^-1 - 10^-7 P), P(0) = 5000, where t is measured in months. What is the limiting value of the population? At what time will the pop be equal to 1/2 of this limiting value?
//What I know:
a = 10^-1
b = 10^-7
P(0) = 5,000 //initial value
//Answer 1
Limiting value = a/b = 10^-1 / 10^-7 = 1,000,000
1/2 the limiting value is 500,000.
//Plug into the logistic equation
500,000 = 10^-1 * 5,000 / (10^-7 + (10^-1 - 10^-7 * 5,000)e^-t10^-1 )
500,000 = 500 / (0.0005 + 0.0995e^-t/10 )
0.005 + 0.0995e^-t/10 = 0.001
9,9885e^-t/10 = 0.0005
e^-t/10 = 0.0050251
-t/10 = ln(0.0050251)
t = 52.93 months //the answer
I get 52.93 months, but the book tells me the answer is 5.29 months. Somehow, my decimal point is wrong. Please tell me where I made my mistake.