Madelyne M.

asked • 08/31/14

What multiplicative rate of change should Hal use in his function?

Hal is asked to write an exponential function to represent the value of a $10,000 investment decreasing at 2% annually.

Dattaprabhakar G.

Madelyne:
 
I am sorry I misread your problem.  The investment IS decreasing at 2% annual rate.  From this we must find the exponential rate of decrease.
 
I will rework the problem and get back to you ASAP.
 
SORRY, once again. I should have been more careful in reading the problem.  Please excuse me.
 
Dattaprabhkar (Dr. G.)
 
 
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08/31/14

Dattaprabhakar G.

Madelyn:
 
Here is the correct answer.
 
Let P(t) be the amount of investment at time t.  We are given that the investment IS decreasing at 2% annual rate.  That means after the first year, the decrease in $10000 will be 10000(-0.02) = $200. So the amount left after the first year is 10000 - 200 = 9800.
 
For exponential growth or decrease there is a general formula given by P(t) = P(0) [exp(rt)]
where,
P(t) is the investment remaining at time t,
P(0) is the investment at time 0, initial investment and
r is the exponential rate of decrease.
 
We have, with t = 1 (after 1 year), P(1) = 9800, P(O) = 10000.  Substituting in the formula, we get
 
9800 = 10000[exp(r)],    Note r times 1 is r.
 
That is 0.98 = exp(r),
so
r = loge (0.98)  = ln (0.98) = -0.0202, to five decimals.  Hence the formula for the exponential function Hal is asked to write is
 
P(t) = P(0)[exp(-0.0202t)].
 
I hope I have not put you to any inconvenience.  If I have, I am sorry.  Please post a comment if you have any questions.
 
Dattaprabhakar (Dr. G.)
 
P.S.  Did you know that to find ln(x) or exp(x), , say x = 0.8, you go to google and in the search area, put exp(0.8) and hit enter.  You will get 2.22554.  Similarly for ln(x).
 
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08/31/14

Dattaprabhakar G.

Madelyne:
 
I just (5 minutes back) posted as a comment, complete solution of your problem!  But I do not find it posted in your space as a comment.
 
Strange!
 
I will rewrite it and send it, but this time I will make a copy and save it on my computer.
 
Dattaprabhakar(Dr. G.)
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08/31/14

Dattaprabhakar G.

Madelyne:
 

There is a general formula for exponential growth or decrease. It is given by

P(t) = P(0)[exp(rt)]

where P(t) is the investment remaining at time t, P(0) is the investment at time 0 (initial), and r is the exponential rate of decrease. We have been given that in the first year (t = 1)the investment decreases by 2%, that is, it reduces from P(0) = 10000 by 10000 x 0.02 = 200, to 10000 – 200 = 9800 = P(1). We have

9800 = 10000[exp(r)]                            (r times 1 = r)
so that

0.98 = exp(r), r =loge (0.98) = ln (0.98) = -0.0202 (to 4 decimals) (from google-search)

Now that we have calculated the exponential rate of decrease, the exponential function Hal was asked to write (the general formula) becomes

P(t) = 10000[exp(-0.0202 t)].

Do you know that if you google-search exp(0.8) you get 2.22554, and similarly for ln(specified number)
 
Please post a comment if you have any questions.  I hope I have not inconvenienced you in any way. If so, sorry.
 
Dattaprabhakar (Dr. G.)
Irvine, CA
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08/31/14

Dattaprabhakar G.

Madelyne:
 

There is a general formula for exponetial growth or decrease. It is given by

P(t) = P(0)[exp(rt)]

where P(t) is the investment remaining at time t, P(0) is the investment at time 0 (initial), and r is the exponential rate of decrease. We have been given that in the first year (t = 1)the investment decreases by 2%, that is, it reduces from P(0) = 10000 by 10000 x 0.02 = 200, to 10000 – 200 = 9800 = P(1). We have

9800 = 10000[exp(r)] (r times 1 = r)
so that

0.98 = exp(r), r =loge (0.98) = ln (0.98) = -0.0202 (to 4 decimals)

Now that we have calculated the exponential rate of decrease, the exponential function Hal was asked to write (the general formula) becomes

P(t) = 10000[exp(-0.0202 t)].
Please post a comment if you have any questions.
 
Dattaprabhakar (Dr. G.)
Report

08/31/14

Dattaprabhakar G.

Madelyne:
 
Correct answer:
 
The formula for exponential growth or decrease is given by

P(t) = P(0)[exp(rt)]

where P(t) is the investment remaining at time t, P(0) is the investment at time 0 (initial), and r is the exponential rate of decrease. We have been given that in the first year (t = 1)the investment decreases by 2%, that is, it reduces from P(0) = 10000 by 10000 x 0.02 = 200, to 10000 – 200 = 9800 = P(1). We have

9800 = 10000[exp(r)] (r times 1 = r)

so that

0.98 = exp(r), r =loge (0.98) = ln (0.98) = -0.0202 (to 4 decimals) 

Now that we have calculated the exponential rate of decrease, the exponential function Hal was asked to write becomes

P(t) = 10000[exp(-0.0202 t)].

Please post a comment if you have any questions.
 
Dattaprabhakar (Dr. G.)
Irvine, CA
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08/31/14

2 Answers By Expert Tutors

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Kyle L. answered • 08/07/25

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Hello, thank you for taking the time to post your question!


The multiplicative rate of change is going to be based on the percentage decrease of 2% … so you want to take

100% - 2% = 98% = 0.98 in decimal terms


So 0.98 is the multiplicative rate of change for this situation


I hope that helps get you moving in the right direction! Feel free to reach out if you have any additional questions beyond that :)

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Dattaprabhakar G. answered • 08/31/14

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Madelyne:
 
Correct answer:

The general formula for exponential growth or decrease is given by

P(t) = P(0)[exp(rt)]

where P(t) is the investment remaining at time t, P(0) is the investment at time 0 (initial), and r is the exponential rate of decrease. We have been given that in the first year (t = 1)the investment decreases by 2%, that is, it reduces from P(0) = 10000 by 10000 x 0.02 = 200, to 10000 – 200 = 9800 = P(1). We have

9800 = 10000[exp(r)] (r times 1 = r)

so that

0.98 = exp(r), r =loge (0.98) = ln (0.98) = -0.0202 (to 4 decimals)

Now that we have calculated the exponential rate of decrease, the exponential function Hal was asked to write becomes

P(t) = 10000[exp(-0.0202 t)].

Please post a comment if you receive this and also if you have any questions.

Datttaprabhkar (Dr. G.)
Irvine, CA
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