Question

A biologist took a count of the number of fish in a particular lake, and recounted the lake’s population of fish on each of the next six weeks.

Find a quadratic function that models the data as a function of x, the number of weeks. Use the model to estimate the number of fish at the lake on week 8.


P(x) = 13x^2 – 10x + 350; 917 fish
P(x) = 13x^2 – 10x + 350; 1,102 fish
P(x) = 18x^2 + 10x + 300; 1,252 fish
P(x) = 18x^2 + 10x + 300; 1,532 fish

Answer 1

Week
0
1
2
3
4
5
6

Answer 2

Population
350
353
382
437
518
625
758

Answer 3

first, in the given options
what is the population in week 0 in each of them ?

Answer 4

Wouldnt I write it like this : 13x^2 – 10x + 350;
13(0)^2-10(350)+350 ?

Answer 5

for the first : 13x^2 – 10x + 350
plugging x = 0
13 * (0^2) - 10 * 0 + 350 = .. ?

Answer 6

You lost me . . . :/

Answer 7

in order to get from each equation
the amount of fish every week
we plug x = number of week

so in order to get the number of fish in week number 0 we just replace x by 0 in each equation

the ones that will be good for us will be the ones with 350 fish in week number 0.

Answer 8

Oh , i see that . . .

Answer 9

so what can you tell about the number of fish in week number 0 according to each one of the equations ?

Answer 10

I dont understand what you are asking ?

Answer 11

now in order to see which of the equations really shows 350 fish in week 0
we need to plug x =0 in each of them
so now we have to check it

Answer 12

plug 0 into what equation though ?

Answer 13

into each one of them
so we know which of them shows 350 fish in week number 0..

Answer 14

so it would look like :
13(0)^2 – 10(0) + 350 ?

Answer 15

yes so what is left ?

Answer 16

&& then the second one would look like :
13(2)^2 – 10(2) + 350

Answer 17

it would be just 350 ?

Answer 18

yes!.. so in the first we have 350
what in the second ? third ? last ?

Answer 19

the second i got 302 ?

Answer 20

the second looks exactly as the first so i dont understand how you got 302

Answer 21

am i supposed to plug in 0 for the second one too ?

Answer 22

yes.. no we are checking which of the equations shows 350 fish for week number 0
so we will be able to eliminate the ones who shows different number

Answer 23

do you understand the logic behind this process ?
in the given data we had 350 fish in week number 0
so our equation should show this thing as well !

Answer 24

because we are looking for an equation that represents the data

Answer 25

I kinda understand it .

Answer 26

so the 2nd would be:
13(0)^2 – 10(0) + 350

Answer 27

its the same exact thing as the 1st ?

Answer 28

look at the equations .. the same thing

Answer 29

Yea , i see .,

Answer 30

what about the third ? and the last ?

Answer 31

just calculate and give the answer .. you dont need to show me that you plug 0 in there

Answer 32

so the 3rd would be :
18(0)^2 + 10(0) + 300
=300 ?

Answer 33

correct

Answer 34

So , now what ? lol

Answer 35

what about the last ?

Answer 36

its the same thing as the 3rd .

Answer 37

good .. so what two equations are good for us so far ?

Answer 38

1st ?

Answer 39

1 and 2 .. they are the same so far

Answer 40

Yes .

Answer 41

and both of them passed our first "test" - what is the number of fish in week number 0

Answer 42

350 ?

Answer 43

it wasnt a question .. i just said

Answer 44

ohh lol .

Answer 45

i feel slow .

Answer 46

so now we left with

P(x) = 13x^2 – 10x + 350; 917 fish
P(x) = 13x^2 – 10x + 350; 1,102 fish

the difference between those two answers
is " Use the model to estimate the number of fish at the lake on week 8."

now we have to plug 8 in the equation
and see which of them correct
is in week number 8 we have 917 fish or 1,102 fish

Answer 47

So , it will be :
13(8)^2-10(8)+350
=1102

Answer 48

its the 2nd one (;

Answer 49

good job

Answer 50

i really hope you understand what we did

Answer 51

Thank you , couldn't have done it without you thoughh ! (;

Answer 52

Yea I did . I understand it now , promise !

Answer 53

sometimes the process is long and we forget what we are actually looking for

Answer 54

the first step was eliminating some of the equations using the given data
then the second step was checking the validity of the data given in the answer itself

Answer 55

Yep . . . I see .