Question 1207489: An insect population grows every day. The growth rate of the number of insects per day is described by the function i(t) = 100. e^0.2t, where t is the number of days counted from a certain moment. Calculate using ICT but NOTE the calculation that you make with ICT!
a) By how many insects has the population grown during the first ten days?
• Calculation:
Answer:
b) By how many insects has the population grown on the eleventh day?
• Calculation:
Answer:
Found 4 solutions by Theo, josgarithmetic, ikleyn, greenestamps:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! your equation appears to be:
f(t) = 100 * e ^ (.2 * t).
when t = 0, the formula becomes f(t) = 100 * e ^ (0).
since e ^ (.2 * 0) = 1, then f(t) = 100 when t = 0.
it appears that, at t = 0, you have 100 insects.
when t = 10, you have 100 * e ^ (.2 * 10) = 738.9056099 insects.
that is 638.9056099 more than when you started.
that's your growth.
when t = 11, you will have 738.9056099 * e ^ (.2 * (11 - 10) = 738.9056099 * e ^ (.2 * 1) = 902.5013499 insects.
your growth is 902.5013499 minus 738.9056099 = 163.59574 insects in 1 additional day from day 10 to day 11.
this equation can be graphed.
it looks like this.
since the number of insects has to be an integer, then you need to round your results.
your choice is to round your final results only, or to round all intermediate results as well.
an argument can be made that you would need to round all intermediate results, since the number of insects has to be an integer every time you take a tally.
the spreadsheet below shows the results of not rounding, rounding after all calculations have been performed, rounding down only, rounding up only, or rounding to the nearest integer.
you just need to decide which rounding option is appropriate for the problem you are working n.
Answer by josgarithmetic(39617)  (Show Source):
Answer by ikleyn(52787)  (Show Source):
You can put this solution on YOUR website! .
Hello, in this my post, I will not make calculations for you, using ICT, because I don't know what ICT is.
And, in general, it is out of my principles of tutoring to make somebodies' job, which is pressing buttons,
since it has no any relation to teaching. Making students' job, substituting students in pressing
buttons is the worst possible tutors service which I can imagine - so, I never do it.
My post is about another issue. I simply want to explain you
what your problem is about and, conceptually, what you should do with it.
If the function i(t) = 100.e((0.2t) is the growth rate, as it is written in your post,
it literally means that the population function is the integral of it, i.e. the anti-derivative of this function.
Anti-derivative of i(t) = 100.e^(0.2t) is N(t) = 500.e^(0.2t) (plus, formally, a constant).
It means that calculations in the posts of other tutors (both @josgatithmetic and @Theo) are INCORRECT and IRRELEVANT.
They are irrelevant, since they MISTAKENLY treat the given growth rate function i(t) as a population function N(t).
So, again, the calculations in their posts are not consistent with wording in your post.
This discrepancy is that fundamental danger moment which I want to point/(to warn)/(to aware) you.
Next, to answer first question, you have two options, or two ways.
First way is to calculate 10 values i(1), i(2), i(3), . . . , i(10) separately and then add all of them.
You may round each value on the way, or round the final sum
- I do not think that it makes any serious difference.
Notice that the sequence i(1), i(2), i(3), . . . , i(10) is a geometric progression,
so you can use the formula for the sum of geometric progression.
Second way is to calculate N(10) - N(0), using the population function.
To answer second question, simply calculate i(11) and round it.
-----------------
In his post, regarding my writing, @greenestamps positions himself as a person,
who can read in the head of visitors. Perhaps, he has such super-skills, but I don't.
Moreover, I am even not going to do it.
I react as I read, and I read what I see. My reaction was perfectly adequate to the posted problem.
/\/\/\/\/\/\/\/
I re-read the incoming post again. I see there this written text
"The growth rate of the number of insects per day is described by the function i(t) = 100. e^0.2t".
OK, I may assume that the person made a typo or a mistake saying "the growth rate" instead of "population",
but next he writes "per day", highlighting the idea and clearly showing that it is really "the growth rate"
and is not a typo or mistake.
So, I really do not understand why I should treat it in another way.
Only because @greenestamps wants it ?
Resume: My solution is ABSOLUTELY CORRECT * * * * * Comment from @greenestamps is ANSOLUTELY WRONG.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Don't ignore the good responses from tutors @josgarithmetic and @Theo, as suggested in the response from tutor @ikleyn.
The expression "100. e^0.2t" is in the standard form for a population function.
While it is awkward to state that "The growth rate... is described by..." that function, clearly the intent is that it is the population function and not the population growth rate function.
Literally, the statement of the problem says that the growth rate is DESCRIBED BY the given function; it does not say that the given function IS the growth rate function.
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