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? milliliters

Answer 1

To model the amount of medication remaining in Martha's body, we can use the exponential decay formula in the form \( y = a(b)^x \).

Here, we have:

- The initial amount of medication, \( a = 30 \) milliliters.
- The decay factor, \( b \). Given that the amount of medication decreases by \( \frac{1}{5} \) each hour, we can interpret that as \( 4/5 \) of the medication remains after each hour. Therefore, \( b = \frac{4}{5} \).

With these values, we can write the equation:

\[
y = 30 \left(\frac{4}{5}\right)^x
\]

Now, we need to find out how much medication will remain in Martha's body after \( x = 6 \) hours:

\[
y = 30 \left(\frac{4}{5}\right)^6
\]

Calculating \( \left(\frac{4}{5}\right)^6 \):

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

Now multiply this by 30:

\[
y = 30 \cdot \frac{4096}{15625} = \frac{30 \times 4096}{15625} = \frac{122880}{15625}
\]

Now we compute this value:

Using a calculator or performing the division, we find:

\[
\frac{122880}{15625} \approx 7.86
\]

Rounding to the nearest milliliter:

\[
y \approx 8 \text{ milliliters}
\]

Thus, the amount of medication remaining in Martha's body after 6 hours is approximately **8 milliliters**.