The amount of medication in an individual's bloodstream decreases by 10% every hour.

Suppose a patient was given 170 milligrams of the medication. Use this information to complete the table below.

Hours Passed, t	Medication Remaining, mg
0	170
1
Number

2
Number

3
Number

4
Number

Round to three decimal places for each hour, as needed.

Using the general formula,  y=p0(1+r)t
as a guide, write the exponential equation that models this situation.

Where y is the amount of medication remaining, and t is the hours passed.

y =
Number
(
Number
)  t

Use the equation you wrote to estimate the amount of medication remaining after 7 hours.

Round to three decimal places, as needed.

There will be
Number
mg of medication remaining after 7 hours.

### 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.

The amount of medication in an individual's bloodstream decreases by 10% every hour. Suppose a patient was given 170 milligrams of the medication. Use this information to complete the table below. Hours Passed, t: 0 (170 mg), 1, 2, 3, 4. Round to three decimal places for each hour. Write the exponential equation that models the situation. Estimate the amount of medication remaining after 7 hours.

Explore the exponential decay of medication in the bloodstream with a 170 mg initial dose and 10% decrease per hour. Learn how to model this scenario with the formula y = 170(0.90)^t and estimate the medication remaining after several hours. Ideal for Grades 9-12 students.

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