Question

The half life of phospherous 32 is about 14 days. There are 6.6 grMA INITIALLY.
Express the amount of phospherous 32 remaining as a function of time (t).
We have to use the equation y=ae^-kt

Answer 1

\[A(t)=6.6\times \left(\frac{1}{2}\right)^{\frac{t}{14}}\] will work, without much effort

Answer 2

when i would solve the equation i would get a very small numbe. Ok i had that set up wrong

Answer 3

your teacher won't like it though, they want you to use base \(e\) and solve for \(k\) first

Answer 4

we can do it that way too if you like, i am sure that is what you are supposed to do

Answer 5

So how would i exactly get that t/14 by itself?

Answer 6

you know the half life is \(14\) days you the first step is to solve
\[e^{14t}=\frac{1}{2}\] for \(t\)

Answer 7

oh i have confused you
my initial answer \(A(t)=6.6\times \left(\frac{1}{2}\right)^{\frac{t}{14}}\) is also my final answer
it is an alternative to \(A_0e^{kt}\)

Answer 8

ok so i just have to ln both sides to cancel e

Answer 9

lets do it the way your teacher wants

Answer 10

solve
\[e^{14k}=\frac{1}{2}\] because you know it 14 days, i.e. when \(t=14\) you have half of what you started with

Answer 11

solve quickly
\[e^{14k}=.5\]
\[14k=\ln(.5)\]
\[k=\frac{\ln(.5)}{14}\]

Answer 12

Ok i was multiplying by taht 14

Answer 13

you can easily generalize this
now we know \(k=-0.0495\) rounded

Answer 14

and your final answer is
\[A(t)=6.6e^{-.0495t}\]

Answer 15

ok so the i put14 in for t and taht answer in for k and solve the equation?

Answer 16

hold on, i sense you are maybe confused, so lets go slow

Answer 17

ya im not really sure how to solv t

Answer 18

this is the question
"Express the amount of phospherous 32 remaining as a function of time (t). We have to use the equation \(y=ae^{-kt}\)"

Answer 19

you are not asked to solve for \(t\)
you are asked to find the function that gives the amount at time \(t\)
the function is a function of \(t\) so you need a \(t\) in the answer

Answer 20

your job was to take the given information and solve for \(\huge k\)

Answer 21

one you have it, you are done, you replace \(k\) by your answer in the function
\[y=ae^{kt}\] and you are finished

Answer 22

you solve for \(t\) when you are asked something like "how long before..." or "how old is..."
that is when you want the time

Answer 23

ok i i got that i already know t k is my unknown. Thanks

Answer 24

well you do not "know" \(t\) as \(t\) is variable, but yes, you need to solve for \(k\) to get the function
i hope the steps to find it were clear
ignore my first answer, i think it confused the issue

Answer 25

Ok thank you

Answer 26

yw