Verify that the following function is a probability mass function, and determine the requested probabilities.
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Answer:
Step-by-step explanation:
You need to find the model for this situation using the points given. If you are given 13 mg immediately, the coordinate point for that is (0, 13), where the x coordinate is the time in minutes and the y coordinate is the amount of dye in mg.
If after 12 minutes, 4.75 mg remain, the coordinate point for that is (12, 4.75). Just so you know, the time unit always goes in for x, never for y.
Now that we have these coordinates we are going to use them as a system and find the model. Begin with the point (0, 13) in
where A is the amount left in the body, a is the initial amount injected, b is the rate at which it decays, and t is the time in minutes. Plug in the first set of coordinates:
and since anything raised to the power of 0 is equal to 1, we have
13 = a. Now we can use that and write another equation with the other set of coordinates to solve for b:
and divide both sides by 13 (don't round anything yet) to get
.3653846154 = b¹². Now take the 12th root of both sides (you need a scientific calculator for this) to get that
b = .9195228407 Yikes.
Now for the model, using the fact that a = 13 and b = .9195228407:
We are told that the amount that we are allowed to leave with still remaining in our system cannot be more than 2 mg, so we sub in a 2 for A:
and begin by dividing away 13 to get
Take the natural log of both sides, which allows us to bring the t out front:
ln(.1538461538) = t ln(.9195228407) and finally divide by ln(.9195228407) to get
t = 22.3 minutes