Step 1: Constructing the general exponential decay equation

The general exponential decay formula is:

$y = p_0 (1 + r)^t$

Where:

• $y$ is the remaining amount of medication after $t$ hours,
• $p_0$ is the initial amount of medication (170 mg),
• $r$ is the rate of change (in this case, a 10% decrease, which means $r = -0.10$),
• $t$ is the number of hours passed.

Thus, the formula for this scenario becomes:

$y = 170(1 - 0.10)^t = 170(0.90)^t$

Step 2: Filling in the table

Now, we can use this equation to calculate the remaining amount of medication for each hour.

For $t = 1$:

$y = 170(0.90)^1 = 170(0.90) = 153 \, \text{mg}$

For $t = 2$:

$y = 170(0.90)^2 = 170(0.81) = 137.7 \, \text{mg}$

For $t = 3$:

$y = 170(0.90)^3 = 170(0.729) = 123.93 \, \text{mg}$

For $t = 4$:

$y = 170(0.90)^4 = 170(0.6561) = 111.537 \, \text{mg}$

Table Summary

Hours Passed (t)	Medication Remaining (mg)
0	170
1	153
2	137.7
3	123.93
4	111.537

Step 3: Estimating the medication remaining after 7 hours

We now use the equation to calculate the remaining amount after 7 hours:

$y = 170(0.90)^7$

Let’s calculate that.

$y \approx 170(0.4782969) = 81.310 \, \text{mg}$

Final Answer:

There will be approximately 81.310 mg of medication remaining after 7 hours.

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Would you like further details or have any questions?

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Related Questions:

1. How would the equation change if the medication decreased by 15% every hour?
2. How long will it take for the medication to drop below 50 mg?
3. What is the half-life of this medication in terms of hours?
4. How much medication remains after 10 hours?
5. What if the initial dose was 200 mg instead of 170 mg?

Tip:

Exponential decay models like this one can be used to predict the behavior of many real-life situations, such as radioactive decay or cooling rates in physics.