$\begingroup$

"A radioactive substance decays according to the formula $W= 20e^{-kt}$ grams where $t$ is the time in hours."

$a)$ Find $k$ given that after $50$ hours the weight is $10$ grams.

$b-d$ I do not need help with

$e)$ Show that $\frac{dW}{dt}$ is proportional to the weight of substance remaining.

For $a)$ I got $k= 0.0139$ (to be exact it was $1/50\cdot\ln2$, but I rounded it to $4$ significant figures)

I am not exactly sure what I am supposed to find in $e)$. When I derive the formula, it is equal to $(1/50\cdot\ln2) \cdot 20e^{-(1/50\cdot\ln2)t} $. What do they want me to find/show? I got the derived equation, is that all I am supposed to show?

$\endgroup$ 2

1 Answer

$\begingroup$

You want to find $\frac{dW}{dt}$ and show that it is a constant times $W$.

Taking the derivative of $W=20e^{-kt}$ using the chain rule we get

$$\frac{dW}{dt}=20e^{-kt}\cdot(-k)=-kW$$

We know that $-k$ is a constant, so we are done.

$\endgroup$ 0

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy