# Application of diffentation

1. May 14, 2014

### delsoo

1. The problem statement, all variables and given/known data
can anyone expalin where to get the equation for max growth? and the value? this is not my working. this is just the sample ans.

2. Relevant equations

3. The attempt at a solution

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2. May 14, 2014

### Simon Bridge

You get it from the equation of the growth.

3. May 14, 2014

### delsoo

which is the equation? so confused now.

4. May 14, 2014

### Simon Bridge

Derived as part of the answer to part (a), used in part (b).
The whole question is about you ability to turn words into equations and vise versa.
Consider: what does "rate of population growth" mean?

Q8. A territory will support a maximum population of $P_0$. Let the ratio of the population $P$ to the maximum population be $p$... the rate of change of this ratio is the product of $p$ and the difference between $p$ and $1$.

(a) write down the differential equation in $p$
(b) show that the growth of the population is greatest when $P=\frac{1}{2}P_0$
(c) the population starts at $\frac{1}{4}P_0$ and reaches $\frac{1}{2}P_0$ in 20 years.
Find the time for the population to reach $\frac{7}{8}P_0$.
(d) find the value of ther maximum growth of the population to 3dp.

5. May 15, 2014

### pasmith

Vital omission corrected.

6. May 15, 2014

### HallsofIvy

Staff Emeritus
Does your text really say "defferential equation"? That makes me wonder about the grammar- and, in particular, whether "maximum growth" means the maximum population or maximum rate of growth of the population. If it is "maximum rate of growth", (b) asks you to "show that the growth of the population is greatest when $P= \frac{1}{2}P_0$". What is the rate of growth then?