Questions: If a substance is injected into the bloodstream, the percent of the maximum dosage that is present at time t is given by y=100(1-e^(-0.35(11-t))), where t is in hours, with 0 ≤ t ≤ 11. In how many hours will the percent reach 50%? The percent will reach 50% in □ hours. (Round to the nearest whole number as needed.)

If a substance is injected into the bloodstream, the percent of the maximum dosage that is present at time t is given by y=100(1-e^(-0.35(11-t))), where t is in hours, with 0 ≤ t ≤ 11. In how many hours will the percent reach 50%?

The percent will reach 50% in □ hours. (Round to the nearest whole number as needed.)

Transcript text: If a substance is injected into the bloodstream, the percent of the maximum dosage that is present at time $t$ is given by $y=100\left(1-e^{-0.35(11-t)}\right)$, where $t$ is in hours, with $0 \leq \mathrm{t} \leq 11$. In how many hours will the percent reach $50 \%$ ? The percent will reach $50 \%$ in $\square$ hours. (Round to the nearest whole number as needed.)

Solution

Solution Steps

To find the time $ t $ when the percent of the maximum dosage reaches 50%, we need to solve the equation $ y = 100\left(1-e^{-0.35(11-t)}\right) $ for $ y = 50 $. This involves isolating $ t $ in the equation and solving for it. We will use Python to perform the calculation and round the result to the nearest whole number.

Step 1: Set Up the Equation

We start with the equation that describes the percent of the maximum dosage present at time $ t $: \[ y = 100\left(1 - e^{-0.35(11 - t)}\right) \] We need to find $ t $ when $ y = 50 $.

Step 2: Substitute and Rearrange

Substituting $ y = 50 $ into the equation gives: \[ 50 = 100\left(1 - e^{-0.35(11 - t)}\right) \] Dividing both sides by 100: \[ 0.5 = 1 - e^{-0.35(11 - t)} \] Rearranging this, we have: \[ e^{-0.35(11 - t)} = 0.5 \]

Step 3: Solve for $ t $

Taking the natural logarithm of both sides: \[ -0.35(11 - t) = \ln(0.5) \] Solving for $ t $: \[ 11 - t = -\frac{\ln(0.5)}{0.35} \] Thus, \[ t = 11 + \frac{\ln(0.5)}{0.35} \]

Step 4: Calculate $ t $

Calculating the value gives: \[ t \approx 12.9804 \] Rounding to the nearest whole number results in: \[ t \approx 13 \]