Question

Please Help Me

Answer 1

Scientists studied two animal populations. Function f(x) = 830(0.8)x models a bear population in a given region x years after the study began. The table models the cougar population in the same region.

Use the table of data to interpret a linear function.

In the year the study began, how many more bears than cougars were in the given region?

Answer 2

@Michele_Laino

Answer 3

a general linear function, between variables \(x,y\), is given by this formula:
\(y=kx+q\) where \(k,q\) are two numeric coefficients, which have to be established using numeric data you provided

Answer 4

for example, here, we can use this function:
\[N = kD + q\]
where \(N\) is the number of cougars, and \(D\) is the number of days

Answer 5

for example, I use the first ordered pair: \((0,790)\), which means \(D=0\) and \(N=790\). Replacing those numbers into my formula above, I get:
\(790=k \cdot 0+q\)
please what is \(q\)?

Answer 6

This my question, if q isn't the number of cougars or days. I don't know what it could me labeled as. Would q be the year?

Answer 7

coefficient \(q\) is the number of cougars at day \(D=0\)
hint: \(k \cdot 0=0\) so I can write:
\(790=0+q\)
therefore: \(q=...?\)

Answer 8

q=790

Answer 9

correct!

Answer 10

Now I update my function, like below:
\(N=kD+790\)

Answer 11

next, I use the second ordered pair \((1,735)\), namely I replace those values into my updated formula, so I get:
\(735= k \cdot 1+790\)
so, what is \(k\)?

Answer 12

k=1

Answer 13

hint:
we have: \(k \cdot 1=k\), so we can write:
\(735=k+790\)

Answer 14

therefore: \(k=735-790=...?\)

Answer 15

oh okay k=-55

Answer 16

so, the updated function, is:
\(N=-55D+790\)

Answer 17

Now, we have to check if that function models your problem

Answer 18

in order to do that, I consider the fourth ordered pair \((6,511)\). Namely, I substitute t6hose values into the updated formula:
left side = \(N=511\)
right side = \(-55 \cdot 6+790=...?\)
please complete

Answer 19

oops.. \(D \) is the number of years, not the number of days

Answer 20

511=-55* 6 +790
511=-330 +790
511=460
1.11

Answer 21

@Michele_Laino just saw your response

Answer 22

please wait, we got right side =460.
Now left side and right side, are different each from other, so our function, doesn't model the population of cougar.
we have to search for another type of function

Answer 23

what function can we use?

Answer 24

The unit that I'm in now is exponential equations and functions, if that helps any

Answer 25

I think about an exponential function, like this:
\[\Large N = q \cdot {k^Y}\]
where \(q,k\) are two numeric coefficients, which they have to be established, \(N,Y\) are number of cougar, and year of study, respectively

Answer 26

I start with the first ordered pair, like before I substitute those values into the new formula, so I get:
\[\Large 790 = q \cdot {k^0}\]
being \(k^0=1\), what is \(q\)?

Answer 27

q= 790

Answer 28

correct!. Now I consider the second ordered pair

Answer 29

q=735

Answer 30

k=1 and q=735

Answer 31

please wait.
I substitute the corresponding values into my updated formula above:
\[\Large 735 = q \cdot {k^1} = 790 \cdot {k^1}\]
what is \(k\)?
hint: \(k^1=k\)

Answer 32

k=1

Answer 33

we have: \(735=790\cdot k\), so:
\(k=735/790=...?\)

Answer 34

k=1.07

Answer 35

what is \(735/790=...?\)

Answer 36

1.07

Answer 37

it is impossible, please retry

Answer 38

.9303

Answer 39

correct!
so the updated function, is:
\[\Large N = 790 \cdot {\left( {0.93} \right)^Y}\]

Answer 40

Now, we have to check if that new function models your problem

Answer 41

in order to do that, again I consider the fourth ordered pair, so I can write this:
left side N=735
right side = \(790 \cdot (0.93)^1=...\)
please complete

Answer 42

735=790 * (0.93)^1
735=734.7

Answer 43

oops.. the fourth ordered pair is another one.
more precisely, we got right side = 734.7, which can be rounded to 735.
Now I consider the third ordered pair, so I can write this:
rleft side: N=683
right side= \(790 \cdot (0.93)^2=...?\)

Answer 44

please complete

Answer 45

683=790 * (0.93)^2
683= 790 * .86
683= 679.4

Answer 46

I got:
right side = 683.27, so we are right.
finally I check my formula on the fifth ordered pair:
left side= 266
right side = \(790 \cdot (0.93)^{15}=...?\)
please complete again

Answer 47

266=790 * (0.93)^15
266=790 * .336
266=265.44

Answer 48

I got:
rigt side =265.99
I think that my formula is correct! Namely the population of cougars can be modeled by this function:
\[\Large N = 790 \cdot {\left( {0.93} \right)^Y}\]

Answer 49

right* side

Answer 50

now, we know that at year \(Y=0\) there was \(790\) cougars

Answer 51

how many bears were at \(Y=0...?\)

Answer 52

it is simple, since we have to replace \(x=0\) into this formula:
\[\Large f\left( x \right){\text{ }} = {\text{ }}830 \cdot {\left( {0.8} \right)^x}\]
so, what is:
\[ \Large {\text{ }}830 \cdot {\left( {0.8} \right)^0=...?}\]

Answer 53

830

Answer 54

correct!, so we have \(830-790=...?\) more bears than cougars at year=0

Answer 55

40

Answer 56

correct!

Answer 57

we have completed your exercise :)

Answer 58

Thank you so much!

Answer 59

:)