Question

Can someone help me answer a couple questions to this picture please :)

Answer 1

@johnweldon1993 @amistre64

Answer 2

@mathstudent55 @whpalmer4

Answer 3

How many fish at the start?

Answer 4

25!

Answer 5

And 1 month later?

Answer 6

50

Answer 7

Right. 50/25 =

One month after there are 50, how many are there? that number / 50 =

see a pattern developing here?

Answer 8

Oh yess they multiply by 2!

Answer 9

exactly. now, how can you write a function that starts out with a value of 25 when x = 0 and doubles each time x increases by 1?

Answer 10

Uhh..so how should I write that. ...f(0) = 0^2?

Answer 11

that's close...

\[f(0) = 25*2^0 = 25*1\]\[f(1) = 25*2^0*2 = 25*2^0*2^1 = 25*2^1\]\[f(2)= 25*2^1*2 = 25*2^2\]\[f(x) = \]

Answer 12

f(3) = 25 * 2^2 = 25 * 2^3

Answer 13

\[f(3) = 25*2^2 * 2 = 25*2^3\]but what is the formula for \(f(x)\) going to be?

Answer 14

hasn't it always been \(f(x) = 25*2^x\) for all the cases we've tried? Remember, \(2^0 = 1\)

Answer 15

Im not sure..its kinds of confusing to me

Answer 16

How can I make it less confusing?

Answer 17

well lets see

Answer 18

I know thaat they multiply 2 each month ...Ok what is the exponential function again?

Answer 19

\[2^x\]

Answer 20

do you know about geometric sequences?

Answer 21

a little bit..

Answer 22

Well, this is a geometric sequence with a common ratio of 2.

Answer 23

Ok I am familiar with this.. OK so can we try it again .the first question is what the exponential function represents in that table

Answer 24

the formula that represents the data in the table is \[P(x) = 25*2^x\]\[P(0) = 25*2^0 = 25*1 = 25\]\[P(1) = 25*2^1 = 25*2 = 50\]\[P(2) = 25*2^2 = 25*4 = 100\]\[P(3) = 25*2^3 = 25*8 = 200\]

Answer 25

Oh okay I see how your doing it noww

Answer 26

Here's a graph of the values of the population

Answer 27

wait how is 25 * 2^0 = 25 * 1

Answer 28

Perhaps more interesting is a semi-log plot where the y-axis plots the logarithm of the population.

Answer 29

Now the points fall in a straight line! That's because exponents and logs are inverse functions of each other...

\[2^2 = 4\]\[2^1=2\]\[\frac{2^2}{2^1}=2^{2-1}= 2^1 = \frac{4}2 = 2\]\[\frac{2^2}{2^2} = 2^{2-2} = 2^0 = \frac{4}{4} = 1\]

Answer 30

\[a^0 = 1, \,a\ne 0\]

Answer 31

So any number (other than 0) to the 0 power = 1.

Answer 32

Ohhhhh ok ok I see

Answer 33

It should make sense — if 10^2 = 10*10 = 100, then dividing that by 10 gives us 100/10 = 10 which is 10^1, and dividing by 10 again gives 1 which is 10^0, dividing by 10 again gives us 0.1 (or 1/10) which is 10^-1, etc.

Answer 34

Are you starting to feel comfortable with our function for the population?

Answer 35

yess I am

Answer 36

Now I have to say what the numbers and variabls represent

Answer 37

Well, what does that 25 represent?

Answer 38

the starting fish ?

Answer 39

yes! the starting population

Answer 40

I dk what p would stand for..they taught me it mean amount of money invested

Answer 41

Population, perhaps? :-)

Answer 42

just because a letter means something in one formula doesn't mean anything about what it means in a formula in an unrelated area...there's nothing magic about the choice of letters for variable names, just an attempt to make the purpose and meaning more apparent.

For example, the Pythagorean theorem is usually written \[a^2+b^2=c^2\]but there's absolutely no reason it couldn't have been \[u^2+v^2=w^2\] or \[f^2+q^2 = z^2\]

Answer 43

but isnt that what 25 is?

Answer 44

No, \(P(t)\) is the population at time \(t\)
25 is the population at t = 0, aka the initial population

Answer 45

I guess we were writing \(P(x)\), not trying to confuse you by switching variable names...

Answer 46

ohhhh ok so they are basically the same thing

Answer 47

50 represents the increase amount after 1 month..correct?

Answer 48

no, P(x) is a function, that gives the population at a certain time. 25 is the initial value of the population, and P(0) = 25.

Answer 49

Where do you see 50 in the formula?

Answer 50

\[P(x) = 25*2^x\]is the formula

Answer 51

oh I thought it wanted everything from the table sorry

Answer 52

Ok the 2^x is the number of times they multiplied with an exponent of x

Answer 53

or 2^x no idea how to describe this one

Answer 54

"what does each number and variable represent in your function from #1?"

2 means doubling, doesn't it? x is the number of months that have passed, so 2^x shows the population growth

Answer 55

lol yess. thanks

Answer 56

im gonna try # 3

Answer 57

wait how is it going to reach 500..if its even numbers im working with..unless i change the number 2

Answer 58

@whpalmer4

Answer 59

Well, you're making the assumption that 500 is reached exactly on a 1 month boundary...and as you've seen, it isn't.

Answer 60

You need to use logarithms to find the answer.

\[500 = 25*2^x\]Divide both sides by 25\]\[20=2^x\]Now we take the logarithm base 2 of each side:\[\log_2 20 = \log_2 2^x\]But \[\log_b b^x = x\]so we have\[\log_2 20 = x\]

Answer 61

Now you probably don't have a button on your calculator that will calculate logs to base 2, but we can convert a logarithm to another base:

\[\log_b x = \frac{\log_a x}{\log_a b}\]So if your calculator does common logarithms (base 10), you could do \[\log_2 20 = \frac{\log_{10} 20}{\log_{10} 2} \approx 4.32\]

Answer 62

and if you look at my second graph, if you go over to x = 4 1/3 and go up to the line that you would have if you connected the points, then go over to the y axis, you'd find yourself at 500.

Answer 63

Wow okayy. lol

Answer 64

So at month 4 it should be 500 fishes?