Question

Martha just took 30 milliliters of medication. She read online that the amount of medication remaining in her body will decrease by about
1

5
each hour.
Write an exponential equation in the form y=a(b)x that can model the amount of medication, y, remaining in Martha's body after x hours.
Use whole numbers, decimals, or simplified fractions for the values of a and b.
y =

To the nearest milliliter, how much medication will remain in Martha's body after 6 hours?

Answer 1

To write an exponential equation in the form \( y = a (b)^x \) for the amount of medication remaining in Martha's body, we can define the following:

- \( a \): the initial amount of medication, which is 30 milliliters in this case.
- \( b \): the decay factor. Since the amount decreases by \( \frac{1}{5} \) each hour, we can interpret that as 4/5 of the medication remaining each hour. This means \( b = 1 - \frac{1}{5} = \frac{4}{5} \).

Now, we can write the equation as:
\[
y = 30 \left( \frac{4}{5} \right)^x
\]

To find the amount of medication remaining after 6 hours, we plug in \( x = 6 \):
\[
y = 30 \left( \frac{4}{5} \right)^6
\]

First, we need to calculate \( \left( \frac{4}{5} \right)^6 \):

\[
\left( \frac{4}{5} \right)^6 = \frac{4^6}{5^6} = \frac{4096}{15625}
\]

Now multiplying by 30:
\[
y = 30 \cdot \frac{4096}{15625} = \frac{122880}{15625} \approx 7.87 \text{ milliliters}
\]

To the nearest milliliter, the amount of medication remaining in Martha's body after 6 hours will be approximately:
\[
\boxed{8} \text{ milliliters}
\]