Asked by mysterychicken
1. Exponential growth follows the formula y=a(1+r)^x and exponential decay follows the formula y=a(1-r)^x..
This seems obviously TRUE
But now I'm having doubts about the decay formula
Is this true or false?
This seems obviously TRUE
But now I'm having doubts about the decay formula
Is this true or false?
Answers
Answered by
Steve
Normally exponential growth is expressed as
f(t) = a e^kt
and exponential decay is
f(t) = a e^-kt
If 0<r<1, then the statement is true, since then we have
1-r < 1 so we have a fraction, and exponential of a fraction is e^-kt for some k.
With no restrictions on r, things get a bit dicier.
I think in your case, say you have a 12% growth rate. Then
f(x) = a(1.12)^x
At 12% decay rate would then mean that 88% is left after each interval, so you would indeed have
f(x) = a(1-.12)^x = a(.88)^x
f(t) = a e^kt
and exponential decay is
f(t) = a e^-kt
If 0<r<1, then the statement is true, since then we have
1-r < 1 so we have a fraction, and exponential of a fraction is e^-kt for some k.
With no restrictions on r, things get a bit dicier.
I think in your case, say you have a 12% growth rate. Then
f(x) = a(1.12)^x
At 12% decay rate would then mean that 88% is left after each interval, so you would indeed have
f(x) = a(1-.12)^x = a(.88)^x