# Math Help - derivative with words!

1. ## derivative with words!

Bacteria in a culture increase at a rate proportional to the number present. An initial population of 200 triples in 10 hours. If this pattern of increase continues unabated, then the approximate number of bacteria after 1 full day is
I took 200 and multiply by 10 then multiply that with 12, but I got the wrong answer.

Two objects in motion from t=0 t=3 seconds have positions X lower case 1 (t)= cos (t^2+1) and x lower case 2(t)= e^t/2t. How many times during the 3 seconds do the objects have the same velocity?
I know you suppose to take the derivative but do something with it.

The velocity of a particle in motion along a line (for t=>0) is v(t)=ln(2-t^2). Find the acceleration when the object is at rest.
I took the derivative, so -2t/2-t^2, but I don't know what to do next.

2. Bacteria in a culture increase at a rate proportional to the number present. An initial population of 200 triples in 10 hours. If this pattern of increase continues unabated, then the approximate number of bacteria after 1 full day is?
take time as measured in hours.

You are told that the population satisfies the differential equation:

$\frac{dp}{dt}=kp$

for some constant $k$.

hence:

$
p(t)=Ae^{kt}
$

Put $t=0$ and we get:

$
p(0)=A=200
$

Also we have:

$p(10)=200e^{10k}=600$

so:

$k=\frac{\ln(3)}{10}$

Now evaluate $p(24)$.

CB