Please explain how to do this. No other explanations have worked. thank you

Question

At the beginning of an experiment, a scientist has 356 grams of radioactive goo. After 240 minutes, her sample has decayed to 11.125 grams.

What is the half-life of the goo in minutes?   

Find a formula for G(t)G(t), the amount of goo remaining at time tt. G(t)=G(t)=   

How many grams of goo will remain after 23 minutes?

Answer 1

There are a few formulas you can use. I'll use this one (although you can use one with the decay constant lambda as well)

G(t) = G₀ * 1/2^t/X

I'm going to use X for the half life here -- usually it's written as t_1/2 but the formatting is a bit weird in this window and the X makes it easier to read.

So G(t) is the amount of gas left at time t;

G₀ is the amount of gas you're starting with,

X is the half life (in other words: time it takes for half the substance to decay)

and t is simply time.

So for the first problem, you want to find t1/2.

So setting it up, you have:

11.125 = 356 * 1/2^(240X)

Divide both sides by 356 and you get

0.03125 = 1/2^(240/x)

Now, since you've got an exponential equation, take the natural log of both sides:

ln(0.03125 = ln (1/2^(240/x))

we can rewrite this as ln (0.03125) = 240/x ln (1/2)

Divide both sides by ln (1/2)

ln(0.03125)/ln(1/2) = 240/x

That conveniently divides to 5.

so 5 = 240/x or 5x = 240.

Half life = 48 minutes

This then answers the second question: G(t) = 356*1/2^t/48

See if you can use this to answer the third part on your own, and feel free to post back here if you're stuck.