Calculus Question

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March 18th 2006, 12:07 PM
frozenflames

Calculus Question

Population grows according to the equation , where k is a constant and t is measured in years. If the population doubles every 10 years, then the value of k is

(a) 0.069 (b) 0.200 (c) 0.301 (d) 3.322 (e) 5.000
March 18th 2006, 10:26 PM
earboth

Quote:

Originally Posted by frozenflames

Population grows according to the equation , where k is a constant and t is measured in years. If the population doubles every 10 years, then the value of k is

(a) 0.069 (b) 0.200 (c) 0.301 (d) 3.322 (e) 5.000

Hello,

the growth of this population can be described by the equation:
$p=p_{0} \cdot 10^{k\cdot t}$

where p is the actual amount, $p_{0}$ is the starting value.

Put in the values you know:
$2 \cdot p_{0}=p_{0} \cdot 10^{k\cdot 10}$
$2 = 10^{k\cdot 10}$
solve for k:
$log(2) = k\cdot 10$
now you have to use a caculator or a logarthmic table.

You'll get $k\approx .030103$
that means that none of the given results seem to be right.

Greetings

EB
March 19th 2006, 12:04 AM
Rich B.

Greetings EB:

I solved for k in the equation P(t) = P_0 e^kt which yields k = 0.1*ln(2) approx= 0.069. As it happens, 0.069 is indeed one of the indicated choices.

Enjoy the day,

Rich B.
March 19th 2006, 12:43 AM
earboth

Quote:

Originally Posted by Rich B.

Greetings EB:

I solved for k in the equation P(t) = P_0 e^kt which yields k = 0.1*ln(2) approx= 0.069. As it happens, 0.069 is indeed one of the indicated choices.

Enjoy the day,

Rich B.

Hello, RichB.

of course you are right - BUT. when frozenflames referred to the equation I was not aware, that the equation was meant which you used. It all depends on the base you choose for solving this problem. I have choosen for 10 and thats why I came up with a "wrong" result.

Have a nice day too.

EB
March 19th 2006, 12:53 AM
CaptainBlack

Quote:

Originally Posted by earboth

Hello, RichB.

of course you are right - BUT. when frozenflames referred to the equation I was not aware, that the equation was meant which you used. It all depends on the base you choose for solving this problem. I have choosen for 10 and thats why I came up with a "wrong" result.

Have a nice day too.

EB

In fact it does not matter (in principle) what you take for $b$
(as long as it is positive :D ) in a growth equation of the form:

$ p(t)=p(0)b^{kt} $

for:

$ p(t)=p(0) c^{\log_c(b)\ kt} $

Here there is a good case for using 2 since we are discussing doubling
times, so here it might be natural to use:

$ p(t)=p(0)2^{kt} $

RonL
March 19th 2006, 01:24 AM
Rich B.

Hi Ron:

Right you are. I was just fishing for a base that would yield a parameter, k, that appears in the multiple choice list.

Enjoy.

Rich