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Situation:
A 40 gram sample of a substance thats used for drug research has a k-value of 0.1476.
N=Noe^-kt
No=initial mass( at time t=0)
N=mass at time t
k=a positive constant that depends on the substance itself and on the units used to measure time
t=time, in days

Find the substance's half-life,in days.Round your answer to the nearest tenth.

Respuesta :

missyvee
N=NOe^-kt
N=mass at time t
NO = initial mass
k= 0.1476
t= time, in days

We are asked to find the half-life, which means you want to find how long it will take for half of the substance to decay/disappear (depending on situation). 

If we are looking for half-life, we can simply set N to half of NO (which we are given a value of 40grams for)
Therefore:
N = 20
NO = 40
plugging these values and the value given for k back into the equation you get:

20 = 40e^-0.1476(t)
We are looking for t, so we have to manipulate the formula to get t by itself on one side of the equation.
We can start by dividing 40 from both sides, and you get:
0.5 = e^-0.1476(t)

We have the exponential function "e".
To get rid of e, we can use natural log (ln)
if e^y=x then ln (x) = y
look back at our equation we can set
0.5 = x
-0.1476(t) = y

Rewriting it in natural log form:
ln (0.5) = -0.1476(t)
Plug in ln (0.5) on a calculator to find its value and we get:
-0.693147 = -0.1476(t)
*Note: normally, getting a negative value would suggest that we did something wrong, because you cannot have a negative value as your t (you cannot have negative days), but because there is a negative on both sides of the equation, they will cancel out in this case. 

The last step is to simply divide both sides by -0.1476
therefore:
T = 4.696119
But it asks you for the answer to the nearest tenth (one place after decimal pt) so 
T (half life) = 4.7 days

Hope that helps :)

Answer:

4.7

Step-by-step explanation:

If we are looking for half-life, we can simply set N to half of NO (which we are given a value of 40grams for)

Therefore:

N = 20

NO = 40

plugging these values and the value given for k back into the equation you get:

20 = 40e^-0.1476(t)

We are looking for t, so we have to manipulate the formula to get t by itself on one side of the equation.

We can start by dividing 40 from both sides, and you get:

0.5 = e^-0.1476(t)

We have the exponential function "e".

To get rid of e, we can use natural log (ln)

if e^y=x then ln (x) = y

look back at our equation we can set

0.5 = x

-0.1476(t) = y

Rewriting it in natural log form:

ln (0.5) = -0.1476(t)

Plug in ln (0.5) on a calculator to find its value and we get:

-0.693147 = -0.1476(t)

*Note: normally, getting a negative value would suggest that we did something wrong, because you cannot have a negative value as your t (you cannot have negative days), but because there is a negative on both sides of the equation, they will cancel out in this case.

The last step is to simply divide both sides by -0.1476

therefore:

T = 4.696119

But it asks you for the answer to the nearest tenth (one place after decimal pt) so

T (half life) = 4.7 days

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