Pseudoephedrine hydrochloride is an allergy medication. The polynomial L(t) = 0.03t4 + 0.4t3 – 7.3t2 + 23.1t, where 0 <= t <= 5, models the level of pseudoephedrine hydrochloride, in milligrams, in the bloodstream of a patient t hours after 30 milligrams of the medicine have been taken.
At what times, to the nearest minute, does the level of pseudoephedrine hydrochloride in the bloodstream reach 12 milligrams?
To find the times when the level of pseudoephedrine hydrochloride in the bloodstream reaches 12 milligrams, we need to solve the equation L(t) = 12.
Given the polynomial L(t) = 0.03t^4 + 0.4t^3 - 7.3t^2 + 23.1t, we substitute L(t) with 12:
0.03t^4 + 0.4t^3 - 7.3t^2 + 23.1t = 12
Now, let's solve this equation. However, since it is a polynomial of degree 4, solving it algebraically might be difficult. We can use numerical methods to approximate the solution.
One common method to solve equations numerically is the Newton-Raphson method. However, this method requires an initial guess for the solution. Since we don't have an idea of the initial guess, we'll use a graphical approach to estimate the solution.
We'll plot the graph of the polynomial L(t) = 0.03t^4 + 0.4t^3 - 7.3t^2 + 23.1t and visually identify the x-values where it intersects with y = 12.
Next, we need to find a suitable graphing tool. There are many online graphing calculators available that can help us plot the graph and find the intersections. One such tool is Desmos, which is user-friendly and free to use.
Let's go to Desmos.com and input the equation L(t) = 0.03t^4 + 0.4t^3 - 7.3t^2 + 23.1t in the graphing calculator. We'll also add the horizontal line y = 12 for reference.
After plotting the graph, we can visually identify the x-values where the graph intersects with y = 12. We can zoom in and use the cursor to get more precise values.
Once we have these x-values, we can convert them to minutes by multiplying by 60, as the time is given in hours. Rounding to the nearest minute will give us the final answers.
Using this graphical approach with Desmos or any other graphing tool, we can determine the times (to the nearest minute) when the level of pseudoephedrine hydrochloride in the bloodstream reaches 12 milligrams.