To determine when the patient's temperature reaches its maximum value and what that maximum temperature is, we need to analyze the given quadratic function [tex]\( T(t) = -0.018 t^2 + 0.4248 t + 98.7 \)[/tex].
1. Finding the time when the temperature is at its maximum:
The function [tex]\( T(t) = -0.018 t^2 + 0.4248 t + 98.7 \)[/tex] represents a parabola that opens downward (because the coefficient of [tex]\( t^2 \)[/tex] is negative). The maximum value of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] occurs at its vertex. The time, [tex]\( t \)[/tex], at which the vertex occurs can be found using the formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
Here, [tex]\( a = -0.018 \)[/tex] and [tex]\( b = 0.4248 \)[/tex]. Substituting these values into the formula, we get:
[tex]\[
t = -\frac{0.4248}{2 \times -0.018} = \frac{0.4248}{0.036} \approx 11.8
\][/tex]
So, the patient's temperature reaches its maximum value approximately 11.8 hours after the illness begins.
2. Finding the maximum temperature:
To find the maximum temperature, we substitute [tex]\( t = 11.8 \)[/tex] back into the original function:
[tex]\[
T(11.8) = -0.018 (11.8)^2 + 0.4248 \times 11.8 + 98.7
\][/tex]
Simplifying this expression, we get:
[tex]\[
T(11.8) \approx 101.2
\][/tex]
Therefore, the patient's maximum temperature during the illness is approximately 101.2 degrees Fahrenheit.
Summarizing the answers:
- The patient's temperature reaches its maximum value at approximately 11.8 hours after the illness begins.
- The patient's maximum temperature during the illness is approximately 101.2 degrees Fahrenheit.
These numerical results are already rounded to 1 decimal place.
