growth and decay - help with proof! (1 Viewer)

[german rock on]

Okay so I've looked through maths in focus, fitzpatrick and other sources and I still don't understand something.

The question is;

The rate of decay of a substance is proportional to the mass present at time "t" years. i.e. dM/dt = -kM

a) Show that M = Moe^-kt satisfies the equation dM/dt = -kM

I got the solution but i just dont UNDERSTAND it.
This is what it says;

a) M = Moe^-kt --> dM/dt = -kMoe^-kt = -kM

I understand how they got from step 1 to 2 (just differentiating exponential), but then BAM from step 2 to 3? Does it mean ur meant to input k = 0 something? Any help much appreciated!

 

[shafqat]

-kMoe^-kt = -kM because M = Moe^-kt. It's a substitution with M.

 

[german rock on]

ah right... wow just wrapped my head around that. So easy, damnit!

Just to clarifiy:

M = Moe^-kt [finding the value of M at a given time]

dM/dt = -kM [this is the RATE]

so therefore, when you differentiate M=Moe^-kt, all that happens is the -k goes out the front, leaving the "M" untouched, making it -kM.

Cheers shafquat.