Find initial value exponential decay

Question

You have a heat of 76 after 4 hours and the heat decays exponentially with time towards the outside which is 0 degrees.

The heat decays by a rate of 4.1 degrees per hour.

Find the initial value

So I tried something like this

A(4) = 76 = A0e^(4.1*4)

And then I solve A0 by dividing both sides with e^(4.1*4)

But that gives a unreasonable answer. Any help here?

When does the heat decay by a rate of 4.1 degrees per hour? – WW1 Jan 17 '22 at 23:17 — WW1, Jan 17 '22 at 23:17

WW1 · Answer 1 · 2022-01-18T22:11:21.337

1

Exponential decay to zero must take the form ... $$f(t) = A e^{-kt} $$ for some positive $k$, so... $$f'(t) = -kA e^{-kt} $$ The information you are given tells you that ...

$$f(4) =76 \text{ and }f'(4)=-4.1 $$ You need to solve for $A$, can you take it from there?

******************* EDIT **************************

First solve for $k$ $$ \frac{f'(4)} {f(4)} =\frac{-4.1} {76} = -k $$ Then use that value of $k$ to solve for $A$ $$A = 76\;e^{4k} $$

edited Jan 18 '22 at 22:11

answered Jan 17 '22 at 23:31

WW1

10,497

I still don't get it, something isn't making sense in my head. So i make k = 4.1 and then solve it? Or do i find k? – Corvo Jan 18 '22 at 11:32
See my edit for more details – WW1 Jan 18 '22 at 22:11

Find initial value exponential decay

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