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andreadesirepen
How do I write a half life function of this info? F(x)=45e^-.89x y=45e^-.89x y/45 = e^(-.89x) in y/45 = In e^(-.89x) in y - in 45 = in e^-.89x in y = in e^-.89x+3.81 in y = -.89x+3.81
HELP! It says it should look like this: F(x)=A(1/2)^x. Where A is the initial amount of the substance and x represents the time for decay. I have A, it´s 25. How do i find x?
if:\[F(x)=45e^{-.89x}\]then the value of F(0) will tell you how much the initial amount is equal to. In this case it would be:\[F(0)=45*e^0=45\]
what you need to find is the value of x to get half this value
i.e. what value of x will give you F(x) = 22.5
So it would be, 45(1/2)^22.5 ?
no, you need to solve the equation:\[22.5=45*e^{-0.89x}\]
this will give you a value for x for which F(x) is equal to half its initial value
i plugged, 45*e^-.89 into my calculator and got 18.5. is that right?
how does that solve the equation I listed above?
you need to rearrange that equation to get an expression for x
I can show you the first few steps...
so, starting with:\[22.5=45*e^{-0.89x}\]we divide both sides by 45 to get:\[0.5=e^{-0.89x}\]then take logs of both sides to get:\[\ln(0.5)=-0.89x\]can you do the rest?
I´m lost sorry, no.
which part are you stuck on?
what to do next
do you understand that you need to find x?
did you follow the reasoning above?
i understand i need to find x, but i dont understand how
If I said:\[12=4x\]then would you be able to find the value of x here?
good, so now use the same principals to solve this for x:\[\ln(0.5)=-0.89x\]
tat does not look right to me - please show me your steps so that I can check where you may have made a mistake
i divided .5 by -.89 or is it the other way around?
you cannot do that - the left-hand-side of the equals sign has \(\ln(0.5)\) not just 0.5
so first find the value for \(\ln(0.5)\), then divide that by -0.89
that looks more like it.
yes - you can double check that you have the right value for x by substituting it into your original equation:\[F(x)=45e^{-.89x}\]and see if you get roughly 22.5 as the answer.
why would my A be 45?
In the function given to you, the value of the function when x=0 is 45.
is that the half life function?
oh mah gaaahh. this is tuff
In general you ay have an exponentially decaying function defined as:\[F(x)=Ae^{-bt}\]
in this function t represents the time
and you can see (I hope) that when t=0 F(0)=A
so A represents the initial amount - i.e. the amount at the start
as time passes, the mount decays and gets less and less
at some point, it decays to half its original value
so, at that point F(t) = A/2
now, since we know that:\[F(t)=Ae^{-bt}\]then we can replace F(t) by A/2 to get:\[\frac{A}{2}=Ae^{-bt}\]divide both sides by A to get:\[\frac{1}{2}=e^{-bt}\]take logs of both sides to get:\[\ln(0.5)=-bt\]divide both sides by -b to get:\[t=-\frac{\ln(0.5)}{b}\]this is what the half life is
it represents the amount of time that has to elapse before the amount of material decays to half its original value
I understand the process but it tells me that it needs to look like this: A(1/2)^x
I don't know where you are getting that from - it makes no sense to me.
The half-life of a substance is the time it takes for half of the substance to decay. The exponential function representing half-life is f(x) = A(1/2) where A is the initial amount of the substance and x represents the time for decay. The half-life of the substance will depend on the initial amount of substance you have. In this activity, you will experience this formula in action.
I had to use pennies in the beggining if that helps, so i thought that was A, i used 25 pennies.
where you wrote: f(x) = A(1/2) that corresponds to what I said above when I said that at its half life: so, at that point F(t) = A/2
what this is saying is that when the material (whatever it may be) reaches its half life, then the value of the function f(x) will be equal to HALF the value of f(0). and we know that f(0) = A so, at its half life, f(x) = A/2
that´s the function? why doesnt it look like f(x) = A(1/2)^x ?
it is saying that the VALUE of the function will equal A/2
e.g. if I had a function defined as:\[f(x)=x^2\]and I asked you to find the value of x for which f(x) = 16, then you would do:\[16=x^2\]therefore:\[x=\sqrt{16}=4\]
so note here that I said you need to find the value of x for which: f(x) = 16
so here we are saying that the VALUE of f(x) should be equal to 16
regardless of how the function itself is defined
So how will the half life equation of my probelm look? like this f(x) = A/2 ?
the HALF LIFE is defined as the point at which the VALUE of the function is equal to half its initial value - i.e. the point at which f(x) = A/2
this is not stating that the FUNCTION definition is f(x) = A/2 instead it is stating that the VALUE of f(x) = A/2 and you need to find a suitable x for this to be true
I need to go now - it is very late here and I need some sleep - hope you understand this concept better now.