SOLUTION: Gallium-67 is used in nuclear medicine to help doctors locate inflammation and chronic infections. The patient is injected with a tracer (trace amount) that includes gallium-67, wh

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Question 1194724: Gallium-67 is used in nuclear medicine to help doctors locate inflammation and chronic infections. The patient is injected with a tracer (trace amount) that includes gallium-67, which collects in areas of inflammation and infection. The gallium-67 emits radiation that a special camera can detect. Gallium-67 has a half-life of 3.26 days.
- Give an exponential equation to represent the percentage of the original gallium-67 after t days.
- Determine the amount of gallium-67 left after 4 days.
-Solve your equation to determine the time it will take for there to be 1% of the original gallium-67.
Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
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Half-life of Gallium-67 is 3.26 days.

A=p%2Ae%5E%28kt%29decay

1%2F2=1%2Ae%5E%283.26k%29


ln%281%2F2%29=ln%281%29%2Bln%28e%5E%283.26k%29%29
-0.693147=3.26k
k=-0.693147%2F3.26
k=-0.21262

Revised decay equation: highlight%28A=pe%5E%28-0.21262t%29%29
time t in days
p original amount
A amount after time t

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


In my experience, scientists like to use an exponential equation using the natural base e to solve problems involving the half life of radioactive substances -- as shown by the other tutor.

As a mathematician, I find it far easier to use the definition of half life directly.

The half life of Gallium-67 is 3.26 days. When time is measured in half lives, t days is t/3.26 half lives. Since 1/2 of the original material is left after one half life, the fraction of the original amount remaining after n half lives is %281%2F2%29%5En

In this example, with the tracer element having a half life of 3.26 days, an equation for the fraction remaining after t days is

ANSWER 1: A=A%280%29%281%2F2%29%5E%28t%2F3.26%29 (converted to a percent)

Plug in t=4 days to get

ANSWER 2: %281%2F2%29%5E%284%2F3.26%29=0.4272 = 42.74 percent

Note that result makes sense; 4 days is a bit more than one half life, so the amount remaining should be a bit less than 50%.

You can find the answer to the last question by solving the equation

%281%2F2%29%5E%28t%2F3.26%29=.01

using logarithms.

But finding an accurate numerical answer will require a calculator; so you might as well just use a graphing calculator to find the intersection of the graphs of y=%281%2F2%29%5E%28t%2F3.26%29 and y=0.01.

ANSWER 3: 21.66 days

Note again this result make sense also. The amount remaining after 6 half lives would be 1/2^6=1/64; the amount remaining after 7 half lives would be 1/2^7=1/128. 1% is 1/100, so the answer should be between 6 and 7 half lives, which is roughly between 19 and 23 days.