Three hours after taking a pain medication, 41% of the medication still remains in the bloodstream. Let m be the 1-hour decay factor.

a. What is the 3-hour decay factor as a numeric value?

What is the 3-hour decay factor in terms of m ?

b. Use your results from part a. to find the 1-hour decay factor.
m=

c. Find the 6-hour decay factor.

d. Find the 15-minute decay factor.

(a) 3-hr factor is 0.41

m^3 = 0.41

(b)m = 0.743

(c) m^6 = 0.41^2 = 0.168

(d) 15min = 1/4 hr, so that is
m^(1/4) = 0.928

To answer these questions, we can use exponential decay formula. The general formula for exponential decay is given by:

A = A₀ * e^(-kt)

where:
- A₀ represents the initial amount
- A represents the amount after time t has passed
- k is the decay constant

In this case, we don't have the exact values for A₀ or k. However, we do have some information about the medication remaining in the bloodstream at different times.

a. We are given that 41% of the medication remains in the bloodstream after 3 hours. Let's denote this by A.

A = A₀ * e^(-k(3))

We know that A₀ is the initial amount, and since we have no information about it, we can treat it as a constant.

Now, let's solve for the decay factor m:

A = A₀ * e^(-3k) [Equation 1]

Since we're given that 41% of the medication remains, we can rewrite Equation 1 as:

0.41A₀ = A₀ * e^(-3k)

Canceling A₀ from both sides of the equation:

0.41 = e^(-3k)

Now we can find the 3-hour decay factor as a numeric value by isolating e^(-3k):

e^(-3k) = 0.41

Taking the natural logarithm of both sides:

ln(e^(-3k)) = ln(0.41)

Since ln(e^x) = x:

-3k = ln(0.41)

Dividing both sides by -3 to solve for k:

k = ln(0.41) / -3

So the 3-hour decay factor as a numeric value is ln(0.41) / -3.

b. Now, for the 1-hour decay factor, let's denote it as m. The decay factor corresponds to the proportion of the medication that remains after each hour.

By substituting t = 1 hour into the exponential decay formula, we get:

A = A₀ * e^(-k)

To find m, we need to compare the amount after 1 hour (A) to the initial amount (A₀):

A = A₀ * m

By comparing both equations, we get:

m = e^(-k)

Substituting the value of k from part a, we find:

m = e^(-ln(0.41) / -3)

Simplifying this expression gives us the 1-hour decay factor as a numeric value.

c. To find the 6-hour decay factor, we can use the same approach as in part b. By substituting t = 6 hours into the exponential decay formula, we get:

A = A₀ * m^6

We can then solve for m by comparing this equation to the exponential decay formula:

m^6 = e^(-k * 6)

Substituting the value of k from part a, we can solve for m to find the 6-hour decay factor.

d. Similarly, to find the 15-minute decay factor, we can use the same approach. By substituting t = 15 minutes into the exponential decay formula, we get:

A = A₀ * m^(15/60)

Solving for m by comparing this equation to the exponential decay formula will give us the 15-minute decay factor.