Alright! Let's go through the steps to answer these questions.
The given function representing the patient's temperature over time is:
[tex]\[ T(t) = -0.022 t^2 + 0.5104 t + 97.3 \][/tex]
This is a quadratic function representing a parabola that opens downwards (since the coefficient of [tex]\( t^2 \)[/tex] is negative).
### Step 1: Finding the Time When the Temperature Reaches Its Maximum
For a quadratic function of the form:
[tex]\[ T(t) = at^2 + bt + c \][/tex]
The time [tex]\( t \)[/tex] at which the maximum (vertex) occurs is given by the formula:
[tex]\[ t = \frac{-b}{2a} \][/tex]
Given values:
[tex]\[ a = -0.022 \][/tex]
[tex]\[ b = 0.5104 \][/tex]
Plug these values into the vertex formula:
[tex]\[ t = \frac{-0.5104}{2(-0.022)} \][/tex]
Calculating the above:
[tex]\[ t = \frac{-0.5104}{-0.044} = 11.6 \][/tex]
So, the patient's temperature reaches its maximum value after 11.6 hours.
### Step 2: Finding the Maximum Temperature During the Illness
To find the maximum temperature, we substitute [tex]\( t = 11.6 \)[/tex] back into the original quadratic equation:
[tex]\[ T(t) = -0.022 t^2 + 0.5104 t + 97.3 \][/tex]
Substituting [tex]\( t = 11.6 \)[/tex]:
[tex]\[ T(11.6) = -0.022 (11.6)^2 + 0.5104 (11.6) + 97.3 \][/tex]
Now, calculate each term:
[tex]\[ -0.022 (11.6)^2 = -2.95792 \][/tex]
[tex]\[ 0.5104 (11.6) = 5.92224 \][/tex]
[tex]\[ 97.3 = 97.3 \][/tex]
Adding these together:
[tex]\[ T(11.6) = -2.95792 + 5.92224 + 97.3 = 100.26432 \][/tex]
Rounding to one decimal place, we get:
[tex]\[ T(11.6) = 100.3 \][/tex]
So, the patient's maximum temperature during the illness is 100.3 degrees Fahrenheit.
### Answers:
- After __11.6__ hours, the patient's temperature reaches its maximum value.
- The patient's maximum temperature during the illness is __100.3__ degrees Fahrenheit.
