Question

If a substance is injected into the​ bloodstream, the percent of the maximum dosage that is present at time t is given by yequals100 left parenthesis 1 minus e Superscript negative 0.34 left parenthesis 12 minus t right parenthesis right parenthesis​, where t is in​ hours, with 0less than or equalstless than or equals12. In how many hours will the percent reach 29​%?
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Part 1
The percent will reach 29​% in
  
enter your response here hours.
​(Round to the nearest whole number as​ needed.)

Answer 1

To find the time \( t \) when the percent of the maximum dosage reaches 29%, we can set up the equation using the given formula:

\[
y = 100 \left( 1 - e^{-0.34 (12 - t)} \right)
\]

We want to find \( t \) when \( y = 29 \):

\[
29 = 100 \left( 1 - e^{-0.34 (12 - t)} \right)
\]

Divide both sides by 100:

\[
0.29 = 1 - e^{-0.34 (12 - t)}
\]

Rearranging gives:

\[
e^{-0.34 (12 - t)} = 1 - 0.29
\]
\[
e^{-0.34 (12 - t)} = 0.71
\]

Now, take the natural logarithm of both sides:

\[
-0.34 (12 - t) = \ln(0.71)
\]

Solve for \( t \):

\[
12 - t = \frac{\ln(0.71)}{-0.34}
\]
\[
t = 12 - \frac{\ln(0.71)}{-0.34}
\]

Calculate \( \ln(0.71) \):

\[
\ln(0.71) \approx -0.342
\]

Now substitute:

\[
t = 12 - \frac{-0.342}{-0.34}
\]
\[
t = 12 - \frac{0.342}{0.34}
\]
\[
t \approx 12 - 1.006 \approx 10.993
\]

Rounding to the nearest whole number gives:

\[
t \approx 11
\]

Thus, the percent will reach 29% in approximately **11 hours**.