SOLUTION: Convert the equation f(t) = 116(0.69)^t to the form f(t) = ae^kt Find a = ? and k = ? Round off to three decimal places

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Convert the equation f(t) = 116(0.69)^t to the form f(t) = ae^kt Find a = ? and k = ? Round off to three decimal places      Log On


   

Question 1204990: Convert the equation f(t) = 116(0.69)^t to the form f(t) = ae^kt
Find a = ? and k = ?
Round off to three decimal places
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28t%29+=+116%280.69%29%5Et to f%28t%29+=+ae%5E%28kt%29

a%2A+e%5E%28k+t%29+=+116%2A+0.69%5Et....=>a=116
116%2A+e%5E%28k+t%29+=+116%2A+0.69%5Et.....take log of both sides
ln%28116+%2Ae%5E%28k+%2At%29%29+=+ln%28116%2A+0.69%5Et%29
ln%28116%29%2Bln%28+e%5E%28k+%2At%29%29+=+ln%28116%29%2Bln%28+0.69%5Et%29
%28k%2A+t%29ln%28+e%29+=+t%2Aln%28+0.69%29......substituting ln%28e%29=1, then dividing by t, we have
k+=+ln%28+0.69%29

a+=+116, k+=+ln%28+0.69%29=-0.371

f%28t%29+=+116e%5E%28-0.371t%29


Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

a = 116 since it's the coefficient out front.
Alternatively, plug in t = 0 to find f(0) = 116 in the first equation and f(0) = a in the second equation. That leads to a = 116.


f%28t%29+=+116%2Ae%5E%28kt%29 rewrites to f%28t%29+=+116%2A%28e%5Ek%29%5Et
The base is e%5Ek
Set this equal to the base 0.69 and isolate k.
e%5Ek+=+0.69
k+=+ln%280.69%29
k+=+-0.371 approximately


Summary
a = 116
k = -0.317 approximately