A Particular pain relieving medicine has a decay rate of 24% an hour. A patient was given a dose of the medicine 5 hours ago and there is currently 74mg of the medicine in the patients bloodstream.

initial dose 291.8529. 25% mark in 5.1 hours

To find the original dose of the medicine, we can use the concept of exponential decay. The formula for exponential decay is:

A = A₀ * e^(kt)

Where:
A is the final amount
A₀ is the initial amount
k is the decay constant
t is the time in hours
e is the base of natural logarithms (approximately 2.71828)

In this case, we are given that the decay rate is 24% per hour, which means the decay constant (k) is -0.24.

Let's solve for the original dose (A₀):

A = A₀ * e^(kt)

74mg = A₀ * e^(-0.24 * 5)

To find A₀, we divide both sides by e^(-0.24 * 5):

A₀ = 74mg / e^(-0.24 * 5)

Calculating this value, the original dose of the medicine was approximately 118.4mg.

Now, let's move on to the second question. We need to determine how long it will take for there to be less than 25% of the original dose left in the bloodstream.

Using the same formula:

A = A₀ * e^(kt)

Let's substitute A as 0.25A₀ (25% of the original dose), and solve for t:

0.25A₀ = A₀ * e^(-0.24t)

To isolate t, divide both sides by A₀ and take the natural logarithm of both sides:

ln(0.25) = -0.24t

Now, solve for t:

t = ln(0.25) / -0.24

Calculating this value, the patient will need to wait approximately 2.6 hours (to the nearest tenth of an hour) for there to be less than 25% of the original dose left in the bloodstream.