Question

In a recent stock market downturn, the value of a $5,000 stock is decreasing at 2.3% per month. This situation can be modeled by the equation A(t) = 5,000(0.977)12t, where A(t) is the final amount and t is time in years. Assuming the trend continues, what is the equivalent annual devaluation rate of this stock (rounded to the nearest tenth of a percent) and what is it worth (rounded to the nearest ten dollars) after 1 year?

A. 24.4% and $3,780.00
B. 75.6% and $3,780.00
C. 27.6% and $1,380.00
D. 72.4% and $3,620.00

Answer 1

do you know the answer? @AihberKhan

Answer 2

Okay. This is the equation that is given to us: \(A(t) = 5,000(0.977){12t}\). All we need to do is substitute the \(1\), in for \(t\). This is because it says "after 1 year".

So when we substitute the \(1\) in for \(t\), our equation should look like: \(A(t) = 5,000(0.977){12}\)

Answer 3

yes and i got 3781

Answer 4

but that would be A or B

Answer 5

and when i have chose A it was wrong but now I'm just confused and scared to choose B

Answer 6

It is more than \(2500\). This means that the annual rate of discount cannot be as much as 50%.

Hmm... that is weird. It should be A.

How about I tag someone else here to help you out, because I got A. Let me tag some people :)

Answer 7

@Michele_Laino

We are confused.... can you please help?

Answer 8

@imqwerty

We are confused.... can you please help?

Answer 9

oh okay thank you because I'm so confused

Answer 10

@mathmale

Answer 11

I think that the formula for \(A(t)\), is:
\[\Large A\left( t \right) = 5000 \cdot {\left( {0.977} \right)^{12t}}\]

Answer 12

yeah and i got 3781

Answer 13

when you put 1 for the t

Answer 14

Yeah thats what I thought @Michele_Laino .... but he wrote it differently so I got confused...

Answer 15

Well the problem wrote it differently.

Answer 16

the requested percentage \(r\), is:
\[\Large r = \frac{{A\left( 0 \right) - A\left( 1 \right)}}{{A\left( 0 \right)}} \cdot 100 = \frac{{5000 - 3782}}{{5000}} \cdot 100 = ...?\]

Answer 17

24.4

Answer 18

yes! That's right!

Answer 19

omg i got it right!!

Answer 20

i got 100%% but why did i get it wrong last time

Answer 21

I don't know

Answer 22

maybe the final result has been rounded in a wrong way

Answer 23

can you guys help me with some other question i have to show my work

Answer 24

ok!

Answer 25

is the function like this one:
\[\Large m\left( t \right) = 100 \cdot {0.5^{235t}}\]?

Answer 26

its A(t)=100(0.5)^t/235

Answer 27

if that makes sense

Answer 28

ok! then I rewrite it:\[\Large m\left( t \right) = 100 \cdot {0.5^{\left( {t/235} \right)}}\]

Answer 29

t/235 is written like a fraction i guess thats the same but yeah

Answer 30

then for part A, we can note this:
\[\Large {0.5^{\left( {t/235} \right)}} = {\left( {\frac{1}{2}} \right)^{\left( {t/235} \right)}} = \frac{1}{{{2^{\left( {t/235} \right)}}}} = {2^{ - \left( {t/235} \right)}}\]

Answer 31

so, after a substitution, we get:
\[\huge m\left( t \right) = 100 \cdot {2^{ - \left( {t/235} \right)}}\]

Answer 32

for part B we have to compute this quantity:

\[\Large m\left( {1000} \right) = 100 \cdot {2^{ - \left( {1000/235} \right)}} = ...{\text{grams}}\]

Answer 33

is it 5.236 grams??

Answer 34

yes! Correct!!

Answer 35

so i just put that for the answer in part B?

Answer 36

and ook thank you!

Answer 37

yes!

Answer 38

:)

Answer 39

okay i got it and can you help me with a couple more questions? :/

Answer 40

The population of a particular city is given by the function P(t) = 12,500(1.04)^4t, where t is time in years and P(t) is the population after t years.

Part A: Examine the function. What is the initial population and the percentage growth rate (rounded to the nearest whole percent)?

Part B: What is the population size (rounded to the nearest whole person) in 10 years?

Part C: Discuss collaboratively: How would the population size be affected if 0 < r < 1? Explain.

Answer 41

hint:
the initial population is given by this quantity:
\[\Large P\left( 0 \right) = 12500 \cdot {1.04^{\left( {4 \cdot 0} \right)}} = ...?\]

Answer 42

yeah the t is the 0

Answer 43

yes!

Answer 44

okay so what do i do for partA

Answer 45

what is \(P(0)=...?\)

Answer 46

so i just write the function and put the 0 in for the T?

Answer 47

yes! The initial population is the value of \(P(t)\), when \(t=0\)

Answer 48

ok :)thats what i put for part A

Answer 49

hint:
\[\Large \begin{gathered}
P\left( 0 \right) = 12500 \cdot {1.04^{\left( {4 \cdot 0} \right)}} = \hfill \\
\hfill \\
= 12500 \cdot {1.04^0} = 12500 \cdot 1 = ...? \hfill \\
\end{gathered} \]

Answer 50

so 12500?

Answer 51

that's right!

Answer 52

since multplying by 1 is the same

Answer 53

so i put that ?

Answer 54

yes! It is the initial population

Answer 55

whereas, the percentage growth rate \(r\), I think it is given by the subsequent ratio:
\[\huge r = \frac{{P\left( {t + 1} \right) - P\left( t \right)}}{{P\left( t \right)}} \cdot 100\]

Answer 56

so whats that for? the percentage growth rate?

Answer 57

yes!

Answer 58

oh okay yeh thats also part of part A

Answer 59

so is it r=100/t

Answer 60

we have to substitute the function \(P(t)\):
\[\huge P\left( t \right) = 12500 \cdot {1.04^{4t}}\]
and after that substitution, we get:
\[\Large \begin{gathered}
r = \frac{{P\left( {t + 1} \right) - P\left( t \right)}}{{P\left( t \right)}} \cdot 100 = \hfill \\
\hfill \\
= \frac{{12500 \cdot {{1.04}^{4\left( {t + 1} \right)}} - 12500 \cdot {{1.04}^{4t}}}}{{12500 \cdot {{1.04}^{4t}}}} \cdot 100 = \hfill \\
\hfill \\
= \frac{{12500 \cdot {{1.04}^{4t}}\left( {{{1.04}^4} - 1} \right)}}{{12500 \cdot {{1.04}^{4t}}}} \cdot 100 = \hfill \\
\hfill \\
= \left( {{{1.04}^4} - 1} \right) \cdot 100 = ...\% \hfill \\
\end{gathered} \]

Answer 61

16.98?

Answer 62

%

Answer 63

yes! we have to round such value, so we get \(r=17 \%\)

Answer 64

oh kook so ill put 17% thank you :')

Answer 65

:)

Answer 66

got it so next is part B

Answer 67

Part B-What is the population size (rounded to the nearest whole person) in 10 years?

Answer 68

here we have to compute this quantity:
\[\huge P\left( {10} \right) = 12500 \cdot {1.04^{4 \times 10}} = ...?\]

Answer 69

is it 6001.27?

Answer 70

I got:
\(P(10)=60,012.75\)

Answer 71

omg how?

Answer 72

we can write this:
\[\Large \begin{gathered}
P\left( {10} \right) = 12500 \cdot {1.04^{4 \times 10}} = \hfill \\
\hfill \\
= 12500 \cdot 4.80 = ...? \hfill \\
\end{gathered} \]

Answer 73

oh jk i got it

Answer 74

yeah i see what i did wrong

Answer 75

then we have to round such value:
\[\Large \begin{gathered}
P\left( {10} \right) = 12500 \cdot {1.04^{4 \times 10}} = \hfill \\
\hfill \\
= 12500 \cdot 4.80 \simeq 60013 \hfill \\
\end{gathered} \]

Answer 76

yeah i just did that

Answer 77

so thats the population size?

Answer 78

yes! It is the population after 10 years

Answer 79

okok yay thank you !

Answer 80

Part C: Discuss collaboratively: How would the population size be affected if 0 < r < 1? Explain.

Answer 81

now i need this one

Answer 82

hint:
if \[\Large 0 < r < 1\]
then we can write:

\[\Large \frac{{P\left( {t + 1} \right) - P\left( t \right)}}{{P\left( t \right)}} < 1\]

Answer 83

or more precisely:
\[\Large 0 < \frac{{P\left( {t + 1} \right) - P\left( t \right)}}{{P\left( t \right)}} < 1\]

Answer 84

so do i have to plug in some number for t right ?

Answer 85

the r is supposed to go in the middle

Answer 86

17%?

Answer 87

I think that we have to give a qualitative answer, namely we have to write a statement

Answer 88

yeah we have to write how the population would be affected

Answer 89

the population size

Answer 90

the condition \(0<r<1\) tells that the population is growing, since:
\[\Large P\left( {t + 1} \right) - P\left( t \right) > 0 \Rightarrow P\left( {t + 1} \right) > P\left( t \right)\]

Answer 91

so its affected because it is increasing in size

Answer 92

and on the other hand such growing is limited since we can write this:
\[\Large \frac{{P\left( {t + 1} \right) - P\left( t \right)}}{{P\left( t \right)}} - 1 < 0 \Rightarrow P\left( {t + 1} \right) < 2P\left( t \right)\]

Answer 93

namely the population at a certain year doesn't exceed the double of the population of the preceding year

Answer 94

so that would be the answer?

Answer 95

that the population at a certain year doesn't exceed the double of the population of the preceding year?

Answer 96

yes! I think so!

Answer 97

kook ill just write it and explain