Answer :
To find when the patient's temperature reaches its maximum and what that maximum temperature is, we need to analyze the given quadratic function:
[tex]\[ T(t) = -0.016t^2 + 0.3936t + 97.9 \][/tex]
This function represents a parabola opening downwards (because the coefficient of [tex]\( t^2 \)[/tex] is negative), which means it has a maximum point.
Step 1: Determine when the temperature is maximum
The maximum point of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found at [tex]\( t = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -0.016 \)[/tex] and [tex]\( b = 0.3936 \)[/tex].
Use the formula to find [tex]\( t \)[/tex]:
[tex]\[
t = -\frac{0.3936}{2 \times -0.016}
\][/tex]
Calculate:
[tex]\[
t = \frac{0.3936}{0.032} \approx 12.3
\][/tex]
So, the patient's temperature reaches its maximum 12.3 hours after the illness begins.
Step 2: Find the maximum temperature
Substitute [tex]\( t = 12.3 \)[/tex] back into the temperature equation to find the maximum temperature [tex]\( T(t) \)[/tex]:
[tex]\[
T(12.3) = -0.016(12.3)^2 + 0.3936(12.3) + 97.9
\][/tex]
Calculate:
1. [tex]\( (12.3)^2 = 151.29 \)[/tex]
2. [tex]\(-0.016 \times 151.29 = -2.42064\)[/tex]
3. [tex]\(0.3936 \times 12.3 = 4.84128\)[/tex]
Add them together:
[tex]\[
T(12.3) = -2.42064 + 4.84128 + 97.9 \approx 100.3
\][/tex]
Therefore, the patient's maximum temperature during the illness is approximately 100.3 degrees Fahrenheit.
Answer Summary:
- The patient's temperature reaches its maximum value 12.3 hours after the illness begins.
- The maximum temperature during the illness is 100.3 degrees Fahrenheit.
[tex]\[ T(t) = -0.016t^2 + 0.3936t + 97.9 \][/tex]
This function represents a parabola opening downwards (because the coefficient of [tex]\( t^2 \)[/tex] is negative), which means it has a maximum point.
Step 1: Determine when the temperature is maximum
The maximum point of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] can be found at [tex]\( t = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -0.016 \)[/tex] and [tex]\( b = 0.3936 \)[/tex].
Use the formula to find [tex]\( t \)[/tex]:
[tex]\[
t = -\frac{0.3936}{2 \times -0.016}
\][/tex]
Calculate:
[tex]\[
t = \frac{0.3936}{0.032} \approx 12.3
\][/tex]
So, the patient's temperature reaches its maximum 12.3 hours after the illness begins.
Step 2: Find the maximum temperature
Substitute [tex]\( t = 12.3 \)[/tex] back into the temperature equation to find the maximum temperature [tex]\( T(t) \)[/tex]:
[tex]\[
T(12.3) = -0.016(12.3)^2 + 0.3936(12.3) + 97.9
\][/tex]
Calculate:
1. [tex]\( (12.3)^2 = 151.29 \)[/tex]
2. [tex]\(-0.016 \times 151.29 = -2.42064\)[/tex]
3. [tex]\(0.3936 \times 12.3 = 4.84128\)[/tex]
Add them together:
[tex]\[
T(12.3) = -2.42064 + 4.84128 + 97.9 \approx 100.3
\][/tex]
Therefore, the patient's maximum temperature during the illness is approximately 100.3 degrees Fahrenheit.
Answer Summary:
- The patient's temperature reaches its maximum value 12.3 hours after the illness begins.
- The maximum temperature during the illness is 100.3 degrees Fahrenheit.
