Final answer:
To find the time when the patient's blood oxygen level was 92%, we set the modeled equation equal to 92 and solved for t using the quadratic formula. So the time is 3 pm.
Step-by-step explanation:
To estimate the time when the patient's blood oxygen level was 92%, we can start by setting the equation given for the blood oxygen level, L = -0.270t² + 3.57t + 80.6, equal to 92%.
So we get the equation:
-0.270t² + 3.57t + 80.6 = 92
First, we'll subtract 92 from both sides to set the equation to zero:
-0.270t² + 3.57t - 11.4 = 0
Now we must solve for t, which represents time. Since the equation is a quadratic, we can either factor it, complete the square, or use the quadratic formula. In this case, factoring may be complex, so the quadratic formula or graphing might be more convenient.
The quadratic formula is:
t = (-b ± √(b² - 4ac)) / (2a)
Where a = -0.270, b = 3.57, and c = -11.4.
After calculating the discriminant (b² - 4ac) and then the two possible solutions for t, we should select the positive value that is closest to a whole number, because negative time does not make sense in this context, and we're estimating to the nearest hour. That will give us the approximate time when the blood oxygen level was 92%.
Finally, you would need to add the value of t (in hours) to the initial time to find the answer. If the initial time was not given, we would assume it to be 0 and just interpret the t value as the time of day on a 24-hour clock (where, for example, t=3 would mean 3 PM).