SOLUTION: Hello, can someone please help me with this problem: Use the properties of exponents to rewrite y=4e^{-0.5t} in the form y=a(1+r)^t or y=a(1-r)^t . Round the value of r to the nea

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Hello, can someone please help me with this problem: Use the properties of exponents to rewrite y=4e^{-0.5t} in the form y=a(1+r)^t or y=a(1-r)^t . Round the value of r to the nea      Log On


   

Question 1155415: Hello, can someone please help me with this problem:
Use the properties of exponents to rewrite y=4e^{-0.5t} in the form y=a(1+r)^t or y=a(1-r)^t . Round the value of r to the nearest thousandth. Then find the percent rate of change to the nearest tenth of percent.
I really need to know what the percent rate of change is.
I have tried but I really just don't understand how to do it.
Please help, Thanks.
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!

y=4e%5E%28-0.5t%29+
in the form y=a%281%2Br%29%5Et or+y=a%281-r%29%5Et+

first you need to find r that makes following statement true

e%5E%28-0.5t%29+=%281%2Br%29%5Et.....take log of both sides

log%28e%5E%28-0.5t%29+%29=log%28%281%2Br%29%5Et%29

%28-0.5t%29log%28e+%29=t%2Alog%28%281%2Br%29%29

%28-0.5cross%28t%29%29log%28e+%29%2Fcross%28t%29=log%28%281%2Br%29%29...write -0.5 as -1%2F2

%28-1%2F2%29log%28e+%29=log%28%281%2Br%29%29

log%28e%5E%28-1%2F2%29+%29=log%28%281%2Br%29%29.....apply exponent rule e%5E%28-1%2F2%29=1%2Fe%5E%281%2F2%29

log%281%2Fe%5E%281%2F2%29+%29=log%28%281%2Br%29%29.......e%5E%281%2F2%29=sqrt%28e%29

log%281%2Fsqrt%28e%29+%29=log%28%281%2Br%29%29.....if log same, we have

1%2Fsqrt%28e%29+=1%2Br

r+=+1%2Fsqrt%28e%29+-+1

your equation in the form y=a%281%2Br%29%5Et is:

a=4, r+=+1%2Fsqrt%28e%29+-+1}

y=4%281%2B1%2Fsqrt%28e%29+-+1%29%5Et

y=4%281%2Fsqrt%28e%29+%29%5Et

check:
+4e%5E%28-0.5t%29=4%281%2Fsqrt%28e%29+%29%5Et+-> true

or
y=4%281-r%29%5Et
y=4%281-%281%2Fsqrt%28e%29+-+1%29%29%5Et
y=4%281-1%2Fsqrt%28e%29%2B+1%29%5Et
y=4%282-1%2Fsqrt%28e%29%29%5Et
check:
+4e%5E%28-0.5t%29=4%282-1%2Fsqrt%28e%29%29%5Et-> not true

so, your answer is: y=4%281%2Fsqrt%28e%29+%29%5Et


Answer by Edwin McCravy(20086) About Me  (Show Source):
You can put this solution on YOUR website!
Her answer is correct, but she used logarithms.  Maybe you aren't supposed to
use logarithms. Maybe you're not studying logarithms yet.  Anyway, here's
how without using logarithms. 

Rewrite y=4e^{-0.5t} in the form y=a(1+r)^t 

   4e%5E%28-0.5t%29+=+a%281%2Br%29%5Et

If those are to be identical, then regardless of what number we choose
to substitute for t, both sides of that equation must be equal.

The easiest number to substitute is 0.

   4e%5E%28-0.5%280%29%29+=+a%281%2Br%29%5E%280%29

   4e%5E0+=+a%281%2Br%29%5E0

Since anything raised to the zero power is 1 [With one exception, the undefined
00, which is of no concern to us here.]

So the above equation becomes:

   4%281%29+=+a%281%29
     4=a

So "a" is 4.  So we substitute 4 for a:

   4e%5E%28-0.5t%29+=+4%281%2Br%29%5Et

We divide both sides by 4

   e%5E%28-0.5t%29+=+%281%2Br%29%5Et

Now we substitute t=1

   e%5E%28-0.5%2A1%29+=+%281%2Br%29%5E1

   e%5E%28-0.5%29+=+%281%2Br%29%5E1

   e%5E%28-0.5%29+=+1%2Br

   e%5E%28-0.5%29-1+=+r

You can rewrite r as

e%5E%28-0.5%29+=+e%5E%28-1%2F2%29+=+1%5E%22%22%2Fe%5E%281%2F2%29+=+1%2Fsqrt%28e%29

which she did.

So to write 

y=4e^{-0.5t} in the form y=a(1+r)^t is to write it

y=4%281%2B%281%2Fsqrt%28e%29-1%29%5E%22%22%29%5Et

And that simplifies to her answer, which is correct but not quite in
the form you asked for.  

Edwin