The label on a prescription bottle directs the patient to take the medicine twice a day. The effective ingredient of the medicine decreases continuously at a rate of 25% per hour. If a dose of medicine containing 1 milligram of the effective ingredient is taken, how much is still present 12 hours later when the second dose is taken?

.75^12

To find out how much of the effective ingredient is still present 12 hours later, we need to understand the rate at which the effective ingredient decreases over time.

Given that the effective ingredient decreases continuously at a rate of 25% per hour, we can first calculate the remaining percentage of the effective ingredient after 12 hours.

To do this, we can use the exponential decay formula:

A = P * (1 - r)^t

Where:
A = Final amount
P = Initial amount
r = Rate of decrease (expressed as a decimal)
t = Time in hours

In this case, the initial amount (P) is 1 milligram, the rate of decrease (r) is 25% or 0.25, and the time (t) is 12 hours.

Using the formula, we can calculate the remaining amount of the effective ingredient after 12 hours:

A = 1 * (1 - 0.25)^12
A = 1 * (0.75)^12
A ≈ 0.063

Therefore, after 12 hours, approximately 0.063 milligrams of the effective ingredient is still present when the second dose is taken.