SOLUTION: the half-life of nitrogen-16 is 7 seconds. how long does it take for 100mg of nitrogen-16 be reduced to 6.25mg?

Question 707616: the half-life of nitrogen-16 is 7 seconds. how long does it take for 100mg of nitrogen-16 be reduced to 6.25mg?
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
This is an exponential growth/decay problem. A general equation for these is:

where
t = number of units of time
A = the amount after t units of time
= the initial amount (IOW: The amount at t = 0)
r = the factor of change for 1 unit of time. If r > 1 then the equation is for growth and if 0 < r < 1 then the equation is for decay.

In your problem:
t = the number of units of 7 seconds that have passed.
A = 6.25
r = 1/2 (since the amount decreases by 1/2 every 7 seconds)
= 100
So your equation is:

Now we solve for t. We start by isolating the base, 1/2, and its exponent by dividing each side by 100:

which simplifies to:

If the left side is a power of 1/2 then we can solve this by hand. If not, then we will need to use exponents. It will easier to see if the left side is a power of 1/2 if we rewrite it as a fraction:

Clearly 25 will go into both 625 and 10000:

Clearly 25 will go again into both 25 and 400:

If we know (or try out) some powers of 1/2 we should quickly find that . So t = 4. Now remember that t is how many sets of 7 seconds that have passed. So the amount reduces from 100 mg to 6.25 mg not in 4 seconds but in 4*7 or 28 seconds.

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