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

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- Mar 18th 2006, 12:07 PMfrozenflamesCalculus 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 - Mar 18th 2006, 10:26 PMearbothQuote:

Originally Posted by**frozenflames**

the growth of this population can be described by the equation:

$\displaystyle p=p_{0} \cdot 10^{k\cdot t}$

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

Put in the values you know:

$\displaystyle 2 \cdot p_{0}=p_{0} \cdot 10^{k\cdot 10}$

$\displaystyle 2 = 10^{k\cdot 10}$

solve for k:

$\displaystyle log(2) = k\cdot 10$

now you have to use a caculator or a logarthmic table.

You'll get $\displaystyle k\approx .030103$

that means that none of the given results seem to be right.

Greetings

EB - Mar 19th 2006, 12:04 AMRich 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. - Mar 19th 2006, 12:43 AMearbothQuote:

Originally Posted by**Rich B.**

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 - Mar 19th 2006, 12:53 AMCaptainBlackQuote:

Originally Posted by**earboth**

(as long as it is positive :D ) in a growth equation of the form:

$\displaystyle

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

$

for:

$\displaystyle

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:

$\displaystyle

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

$

RonL - Mar 19th 2006, 01:24 AMRich 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