Question

I need help with this please! I want someone to go through the steps with me, not tell me the answer. Thanks~! ^_^

Gordon, the financial analyst, has been following two companies as potential investments.

The table below shows the functions to represent the number of sales f(t) of two companies in t years.

Company A B
Sales f(t) = 250(1.05t) f(t) = 175(0.97t)

Gordon is going to compare these companies in three different ways. He is going to compare them by initial sales, by growth or decay, and by the average rate of change from years 1 to 3.

Answer 1

Hey Yato, are these supposed to be exponential functions?\[\large\rm A=250(1.05)^t\qquad\qquad\qquad B=175(0.97)^t\]

Answer 2

Yep!

Answer 3

When t=0,
\(\large\rm 250(1.05)^0=250(1)\)
Hmm ok that's interesting.
I guess this is our initial amount, yes?
When no time has passed, the amount is 250.

When t=1,
\(\large\rm 250(1.05)=262.5\)
hmm that's interesting, the amount increased by 5%.

Looks like the other one is decaying, by 3%
(because 1 - .97 = .03, or to put it another way 100% - 97% = 3%)

Answer 4

Alright, so far it makes sense. So if the exponent is 0 do you substitute it with 1? For the first function.

Answer 5

Yes, anything to the 0th power is 1.

\(\large\rm 7^0=1\)
\(\large\rm \pi^0=1\)
\(\large\rm 1.432^0=1\)

Answer 6

Alright, got it. So can you show me how to find the average rate of change from years 1 to 3 with the same functions? Step by step please. ^_^

Answer 7

I don't like that they called them both f(t).
That's quite confusing.
Instead let's call them A(t) and B(t),\[\large\rm A(t)=250(1.05)^t\qquad\qquad\qquad B(t)=175(0.97)^t\]

Answer 8

Oh wait wait, we're not mixing them together or anything,
so I guess it doesn't really matter XD
whatever, I like A and B,
hopefully you do also lol

Answer 9

Yea, I find it easier that way.

Answer 10

The average change in function A, let's call it \(\large\rm \Delta A\), will be obtained by using our slope formula,\[\large\rm m=\frac{y_2-y_1}{x_2-x_1}\]We're dealing with function notation now though, so it will look a little different,\[\large\rm \Delta A=\frac{A(3)-A(1)}{3-1}\]

A(3) is our second y coordinate,
A(1) is our first.

Answer 11

So we have to do a little work to figure out function A at year 1, and also year 3.

Answer 12

Ok, so how would I do that?

Answer 13

We actually already looked at A(1).
That was the 5% increase we saw from our starting value of 250.

\(\large\rm A(\color{royalblue}{t})=250(1.05)^{\color{royalblue}{t}}\)
To evaluate this function at t=1, we simply replace t with 1,
\(\large\rm A(\color{royalblue}{1})=250(1.05)^{\color{royalblue}{1}}\)
and simplify,
\(\large\rm A(1)=262.5\)

Answer 14

So we would multiply 250 with 1.05 to get 262.5, right?

Answer 15

Yes.
So for A(3), we'll replace t with 3,\[\large\rm A(\color{royalblue}{3})=250(1.05)^{\color{royalblue}{3}}\]

Answer 16

You can put it into a calculator just like you see it,
you don't need the brackets I suppose,
\(\large 250*1.05\text{^}3\)

Answer 17

Ok so 1.05^3 would equal 1.157625, right? And then multiply that with 250, yes?

Answer 18

yes

Answer 19

Ok, I got 289.40625

Answer 20

A(3)=289.40625
A(1)=262.5

Ok great, we have everything we need to calculate the average rate of change.

Average rate of change from years 1 to 3 would be the change in outputs, divided by the change in years,\[\large\rm \Delta A=\frac{289.40625-262.5}{3-1}\]

Answer 21

Ok so the top part would be 26,90625 and the bottom would be 2 and then divide it and get 13,453125, right?

Answer 22

Ok good.
This number is telling us that from year 1 to year 3,
this function A was increasing sales by about 13 and a half per year.

Answer 23

Alright, and then we would do the same thing for function B, right?

Answer 24

Yes, think you can follow those steps and try it on your own? :)
I could check your work when your finished.

Unless it still feels rather confusing.

Answer 25

Alright! I'll try it out myself, I feel a bit more confident. :D

Answer 26

Ok, so far I did \[\frac{ B(3) - B(1) }{ 3 - 1}\] and I solved for B(3). I did:
\[B(3)=175(0.97)^{3}\]
\[B(3)=175(0.912673)\]
\[B(3)=159.717775\]
is that right so far?

Answer 27

yes

Answer 28

Oh wow! I'm actually understanding this a LOT better now! Before you started helping me, I understood nothing. xD I'm going to keep going, please wait for me! >.< Sorry if I take too long.

Answer 29

lol :)

Answer 30

Ok, I did B(1) too.
\[B(1)=175(0.97)^{1}\]
\[B(1)=175(0.97)\]
\[B(1)=169.75\]
and I also wanted to see if I could get how much it was decreasing by too. It's decreasing by 3% right?

Answer 31

Yes.
What you're doing when you multiply 175 by 0.97 is,
you end up with 97% of that 175.

So you've lost 3% of it's original size.

Answer 32

Wow, I'm so happy right now! XD I can't believe I'm getting this right! Let me finish it off now. I hope I get it right! :D

Answer 33

Ok, I think I got it.
\[\frac{ 159.717775 - 169.75 }{ 3 - 1 }\]
\[\frac{ -10.032225 }{ 2 }\]
then I divided and got -5.0161125, is that right?

Answer 34

Ok great job!

And a negative number should make sense in the context of this function B, right? B was decaying over time.

So this value is telling us that over this two year period,
company B was losing an average of 5 sales per year.

Answer 35

Oh my gosh! I can't believe I got it right! You got some magical teaching skills! XD Ok, so would that solve the average rate of change part in the question? Or do we have to subtract the two answers we got to compare them?

Answer 36

Oh, thats a good question :O
It just says "compare" which is fairly ambiguous.
I think what we have is sufficiently what they wanted.

Answer 37

Alright, so can we compare them by initial sales now? And I think we already compared them by growth or decaying, right?

Answer 38

Initial sales should be pretty simple.
The initial amount is the number sitting in front.

Answer 39

Oh alright so would it be 250 for function A and 175 for function B?

Answer 40

Yes.
I don't think they want us to do anything fancy like subtraction.
They didn't ask us to find the difference in the starting amounts.
They just want a simple comparison it sounds like.

Answer 41

Oh ok! So would that be it for this question?

Answer 42

Yaaaa that's probably good.

Yayyy team we did it \c:/

Answer 43

Good job

Answer 44

Yaaaaaayyyy!!! Thank you SOO much! :D

Answer 45

np, keep up the hard work!

Answer 46

I hate to ask this of you because you already helped me SO much with this question but, can you help me with another one please? I'm pulling an all nighter tonight so I can learn as much as I can but, whenever you need to leave, just tell me ok? ^_^