You go to the doctor and he gives you 13 milligrams of radioactive dye. After 20 minutes, 4.5 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
you know that the amount remaining after t minutes is
f(t) = 13 e^(-kt)
Now you know that
13 e^(-20k) = 4.5
k = 0.053, so
f(t) = 13 e^(-0.053t)
Now you want the alarm to sound when p(t) = 2. So, just find t where
13 e^(-0.053t) = 2.0
To determine how long your visit to the doctor will take, we can use the concept of half-life.
The half-life of the radioactive dye is the time it takes for half of the dye to decay. In this case, we know that after 20 minutes, 4.5 milligrams of dye remain in your system. This means that half of the dye has decayed in 20 minutes.
Let's calculate the initial amount of dye you were given by doubling the amount remaining after 20 minutes: 4.5 milligrams * 2 = 9 milligrams.
Now, let's calculate the amount of time it takes for half of the dye to decay. Since 4.5 milligrams remain after 20 minutes, it means that 9 milligrams decayed in 20 minutes, which is one half-life.
We can use the equation for exponential decay:
Amount remaining = Initial amount * (1/2)^(time/half-life)
Substituting the given values:
4.5 milligrams = 9 milligrams * (1/2)^(20 minutes/half-life)
To isolate the half-life, we can divide both sides of the equation by 9 milligrams:
(1/2)^(20 minutes/half-life) = 4.5 milligrams / 9 milligrams
(1/2)^(20 minutes/half-life) = 0.5
Take the logarithm of both sides to solve for the exponent:
log((1/2)^(20 minutes/half-life)) = log(0.5)
(20 minutes/half-life) * log(1/2) = log(0.5)
log(1/2) is approximately -0.30103, so:
(20 minutes/half-life) * (-0.30103) = -0.30103
Solve for half-life:
20 minutes/half-life = -0.30103 / -0.30103
20 minutes/half-life = 1
half-life = 20 minutes
The half-life of the dye is 20 minutes. To reduce the dye to below 2 milligrams, we need to go through at least 6 half-lives (since 2 milligrams is less than half of 4.5 milligrams).
Multiply the half-life by 6 to find the total time needed:
20 minutes * 6 = 120 minutes
Therefore, your visit to the doctor will take approximately 120 minutes (or 2 hours) for the dye to decay to below 2 milligrams.
To find out how long your visit to the doctor will take, we need to figure out how long it takes for the amount of dye in your system to decrease to less than 2 milligrams.
We know that the dye decays at a certain rate over time. Let's call the rate of decay "r" (in milligrams per minute).
The amount of dye remaining in your system after a given time can be modeled using the formula:
R = R0 * e^(-rt)
Where:
- R is the amount of dye remaining in your system after time t,
- R0 is the initial amount of dye (13 milligrams in this case),
- e is the base of the natural logarithm (approximately 2.71828),
- r is the rate of decay, and
- t is the time in minutes.
We know that after 20 minutes, 4.5 milligrams of the dye remain:
4.5 = 13 * e^(-20r)
To determine the value of r, we need to solve this equation. Here's how you do that:
1. Divide both sides of the equation by 13:
4.5/13 = e^(-20r)
2. Take the natural logarithm (ln) of both sides of the equation:
ln(4.5/13) = -20r
3. Divide both sides of the equation by -20:
-ln(4.5/13)/20 = r
Use a calculator to compute the value of r which is approximately -0.081898.
Now that we know the rate of decay, we need to find the time it takes for the amount of dye in your system to decrease to less than 2 milligrams. We can use the same formula and solve for time (t):
2 = 13 * e^(-0.081898t)
To determine the value of t, we follow the same steps as before:
1. Divide both sides of the equation by 13:
2/13 = e^(-0.081898t)
2. Take the natural logarithm of both sides of the equation:
ln(2/13) = -0.081898t
3. Divide both sides of the equation by -0.081898:
-ln(2/13)/0.081898 = t
Use a calculator to compute the value of t, which is approximately 23.9 minutes.
Therefore, your visit to the doctor will take around 23.9 minutes if you were given the dye as soon as you arrived.