Question

You go to the doctor and he gives you 13 milligrams of radioactive dye. After 20 minutes, 4 milligrams of dye remain in your system. To leave the doctor's office, you must pass through a radiation detector without sounding the alarm If the detector will sound the alarm if more than 2 milligrams of the dye are in your system, how long will your visit to the doctor take, assuming you were given the dye as soon as you arrived? Give your answer to the nearest minute. You will spend Preview minutes at the doctor's office, Get help: Video License Points possible: 1 Unlimited attempts. Submit

Answer 1

Answer:

7 minutes

Step-by-step explanation:

At first, I thought this was a hal-life problem, but there was too much left unknown to fill in the half-life equation.  So I went with exponential.

The general form of an exponential equation is

$y=a(b)^x$

where a is the initial amount given, b is the growth/decay rate (ours is decay since the amount of dye is decreasing), y is the amount of dye in your system after x minutes.

We have to come up with a model for this situation before we can answer the question.  The equation has 2 unknowns: a and b.  We need to solve for those.  In order to do that, we need 2 coordinate points to sub in for x and y.  The first coordinate point in (time, amt dye) is (0, 13).  This means that immediately upon arriving at the doctor, after 0 time has gone by, you were given 13 mg of dye.  

The second coordinate point is (20, 4).  This means that after 20 minutes, you had 4 mg of dye in your system.

Use the first coordinate point to solve for a:

$13=a(b)^0$

Since anything in the world raised to a power of 0 is 1, then

a = 13.

Now we will use the second coordinate point along with that value of a to solve for b:

$4=13(b)^{20}$

Begin by dividing both sides by 13 to get

$\frac{4}{13}=b^{20}$

Now you have to undo that power of 20 by taking the 20th root of both sides:

$\sqrt[20]{\frac{4}{13} }=\sqrt[20]{b^{20}}$

The right side simplifies down to just b, and the 20th root of the left side is .9427701685

Therefore, b = .9427701685

The model, then, is:

$y=13(.9427701685)^x$

We need to use that model now to solve for x, the number of minutes it will take for y = 2 mg of dye to remain in your system.  That looks like this:

$2=13(.9427701685)^x$

Begin by dividing both sides by 13 to get

$\frac{2}{13}=.9427701685^x$

We need to get that x out of the position in which it is currently sitting, which is as the exponent.  Do that by taking the natural log of both sides.  That allows us to bring the exponent down out front:

$ln(\frac{2}{13})=x*ln(.9427701685)$

To solve for x, divide both sides by ln(.9427701685).  I have entered the values of the natural logs in the next step:

$\frac{-.4054651081}{-.0589327498}=x$

Divide to get

x = 6.9 or 7 minutes