CPM alg. 2 book unit 7, CF-150 (rabbits and alfalfa)

KichiAiko

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Farmer Ted grows alfalfa and he has been losing more of his crop each month because of the growth of the rabbit population. Ted has been advised to have the rabbits destroyed, but he feels he can simply increase his production to compensate for the alfalfa destroyed by the rabbits. Currently Ted produces 600 tons of alfalfa each month and he plans to increase his production by 5 tons each month. Last month the rabbits destroyed 3 tons of the crop. The rabbit population around Ted's farm is increasing by 15% each month. If Ted allows the rabbits to continue to destroy part of his crop, at what point will they eat everything he produces? Write a system of equations to represent his dilemma and estimate how long it will take.

i've asked my brother for help, but all we've got so far is...

x=months
600+5x=alfalfa
3(.15x)= alfalfa destroyed by rabbits
so, 600+5x=y
3(.15x)=y
but when i combined the two...600+5x=3(.15x) it came out to be 4.55x=-600
x=-131.868
y=-59.341

but then we came up with a different equation which was 605+5x=y
x | y
1 | 605
2 | 1215
3 | 1830
4 | 2450
5 | 3075
 
KichiAiko said:
Farmer Ted grows alfalfa and he has been losing more of his crop each month because of the growth of the rabbit population. Ted has been advised to have the rabbits destroyed, but he feels he can simply increase his production to compensate for the alfalfa destroyed by the rabbits. Currently Ted produces 600 tons of alfalfa each month and he plans to increase his production by 5 tons each month. Last month the rabbits destroyed 3 tons of the crop. The rabbit population around Ted's farm is increasing by 15% each month. If Ted allows the rabbits to continue to destroy part of his crop, at what point will they eat everything he produces?
The growth in the alfalfa is linear: another five tons each month. The growth in the rabbits, however, is exponential: an additional 15% each month over the previous month. So the rabbits will catch up with the alfalfa! :shock:

I'm not sure what equations you came up with...? Try working with the numbers, until you figure out the pattern. You shouldn't have much trouble coming up with the linear equation for the alfalfa in terms of the number of months "m":

. . .alfalfa (tons):
. . . . .m = 0: 600
. . . . .m = 1: 600 + 5
. . . . .m = 2: 600 + 2(5)
. . . . .m = 3: 600 + 3(5)
. . . . .m = 4: 600 + 4(5)

Keep going until you see the pattern. Then do the same thing with the rabbits. Assuming that the amount of damage corresponds to the number of rabbits, the damage is increasing by fifteen percent each month (so each new month is 115% of the previous month):

. . .damage (tons):
. . . . .m = 0: 3
. . . . .m = 1: 1.15(3)
. . . . .m = 2: 1.15(0.15(3)) = 1.15[sup:19iy8b0p]2[/sup:19iy8b0p](3)
. . . . .m = 3: 1.15(0.15[sup:19iy8b0p]2[/sup:19iy8b0p](3)) = 1.15[sup:19iy8b0p]3[/sup:19iy8b0p](3)
. . . . .m = 4: 1.15(0.15[sup:19iy8b0p]3[/sup:19iy8b0p](3)) = 1.15[sup:19iy8b0p]4[/sup:19iy8b0p](3)

Keep going until you see the pattern.

Once you have formulas for the tons grown and the tons damaged after any given number "m" of months, set the formulas equal, and solve for the number of months. :D

Eliz.
 
Hello, KichiAiko!

Farmer Ted grows alfalfa and he has been losing more of his crop each month
because of the growth of the rabbit population.
Ted has been advised to have the rabbits destroyed, but he feels he can simply
increase his production to compensate for the alfalfa destroyed by the rabbits.
Currently Ted produces 600 tons of alfalfa each month and he plans to increase
his production by 5 tons each month. Last month the rabbits destroyed 3 tons of the crop.
The rabbit population around Ted's farm is increasing by 15% each month.
If Ted allows the rabbits to continue to destroy part of his crop,
at what point will they eat everything he produces?
Write a system of equations to represent his dilemma and estimate how long it will take.

\(\displaystyle \text{Last month }(x =0)\text{, Ed produced 600 tons and increases by 5 tons each month.}\)

. . \(\displaystyle \text{In }x\text{ months, his monthly production will be: }\,600 + 5x\text{ tons}\)


\(\displaystyle \text{Last month }(x=0)\text{, the rabbits destroyed 3 tons.}\)
\(\displaystyle \text{Since their population is increasing by 15 percent each month,}\)
. . \(\displaystyle \text{they will destroy 15 percent more each month.}\)

. . \(\displaystyle \text{In }x\text{ months, their amount of destruction will be: }\;3(1.15)^x\text{ tons}\)


\(\displaystyle \text{The question is: when are these two amount equal?}\)

\(\displaystyle \text{We must }estimate\text{ the solution to: }\;600 + 5x \;=\;3(1.15)^x\)


\(\displaystyle \text{We can plot the two functions: }\:\begin{array}{ccc}f(x) & = & 5x + 600 \\ f(x) & = & 3(1.15)^x\end{array}\)

. . \(\displaystyle \text{and let our calculator determine their intersection.}\)


By trial-and-error, I got roughly: \(\displaystyle x \:=\:40\)

In 40 months, the rabbits will totally decimate Ed's alfalfa crop.


\(\displaystyle \text{(It is hoped that, months before, he solved the rabbit problem,}\)
. . \(\displaystyle \text{or he sold the farm and changed careers.)}\)

 
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