Math Help - Logistics formula

1. Logistics formula

The question:

In this exercise we explore further the Pacific sardine population. Studies to fit a logistic model to the Pacific sardine population have yielded the following.

$N= \frac {2.4}{1+239e^{-0.338t}}$

Here, t is measured in years and N is measured in millions of tons of fish. (Round your answers to two decimal places.)

(a) If the current level of the Pacific sardine population is 50,000 tons, how long will it take for the population to recover to the optimum growth level of 1.2 million tons? [Suggestion: One way to solve this is to make a new logistic formula using K = 2.4, r = 0.338, and N(0) = 0.05.]

t = ____ yrs

(I made a new formula and came up with:

$N= \frac {2.4}{1+48e^{-0.338t}}$, but using this is giving me all the wrong answers.)

(b) The value of r used in the model for the Pacific sardine population ignores the effects of fishing. If fishing mortality is taken into account, then r drops to 0.215 per year (with the carrying capacity still at 2.4 million tons). Answer the question in part (a) using this lower value of r.

t = ____ yrs

2. Originally Posted by MathBane
The question:

In this exercise we explore further the Pacific sardine population. Studies to fit a logistic model to the Pacific sardine population have yielded the following.

$N= \frac {2.4}{1+239e^{-0.338t}}$

Here, t is measured in years and N is measured in millions of tons of fish. (Round your answers to two decimal places.)

(a) If the current level of the Pacific sardine population is 50,000 tons, how long will it take for the population to recover to the optimum growth level of 1.2 million tons? [Suggestion: One way to solve this is to make a new logistic formula using K = 2.4, r = 0.338, and N(0) = 0.05.]

t = ____ yrs

(I made a new formula and came up with:

$N= \frac {2.4}{1+48e^{-0.338t}}$, but using this is giving me all the wrong answers.)
If your formula is N(0)= 2.4/(1+ Ae-0.338t then when t= 0, N(0)= 2.4/(1+ A). You want 2.4/(1+ A)= 0.05 so 2.5= (1+ A)(0.05)= 0.05+ 0.05A. Then 2.5- 0.05= 2.45= 0.05A and A= 2.45/0.05= 49, not 48!

(b) The value of r used in the model for the Pacific sardine population ignores the effects of fishing. If fishing mortality is taken into account, then r drops to 0.215 per year (with the carrying capacity still at 2.4 million tons). Answer the question in part (a) using this lower value of r.

t = ____ yrs