Answer :
To solve this problem, we need to determine when the patient's temperature reaches its maximum value and what that maximum temperature is during the illness. The temperature function is given as:
[tex]\[ T(t) = -0.024t^2 + 0.5136t + 97.9 \][/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], and the graph of this function is a downward-opening parabola because the coefficient of [tex]\( t^2 \)[/tex], which is [tex]\(-0.024\)[/tex], is negative.
### Finding the Time of Maximum Temperature
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] gives the maximum point when [tex]\( a < 0 \)[/tex]. The formula to find the time [tex]\( t \)[/tex] at which the maximum temperature occurs is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
For our given function:
- [tex]\( a = -0.024 \)[/tex]
- [tex]\( b = 0.5136 \)[/tex]
Substitute these values into the vertex formula:
[tex]\[ t = -\frac{0.5136}{2 \times -0.024} \][/tex]
Solving this, we find:
[tex]\[ t = 10.7 \][/tex]
So, the patient's temperature reaches its maximum value 10.7 hours after the illness begins.
### Finding the Maximum Temperature
To find the maximum temperature, substitute [tex]\( t = 10.7 \)[/tex] back into the temperature function [tex]\( T(t) \)[/tex]:
[tex]\[ T(10.7) = -0.024 \times (10.7)^2 + 0.5136 \times 10.7 + 97.9 \][/tex]
Calculate this expression to find:
[tex]\[ T(10.7) = 100.6 \][/tex]
Therefore, the patient's maximum temperature during the illness is 100.6 degrees Fahrenheit.
In summary:
- The patient's temperature reaches its maximum 10.7 hours after the illness begins.
- The maximum temperature is 100.6°F.
[tex]\[ T(t) = -0.024t^2 + 0.5136t + 97.9 \][/tex]
This is a quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], and the graph of this function is a downward-opening parabola because the coefficient of [tex]\( t^2 \)[/tex], which is [tex]\(-0.024\)[/tex], is negative.
### Finding the Time of Maximum Temperature
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] gives the maximum point when [tex]\( a < 0 \)[/tex]. The formula to find the time [tex]\( t \)[/tex] at which the maximum temperature occurs is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
For our given function:
- [tex]\( a = -0.024 \)[/tex]
- [tex]\( b = 0.5136 \)[/tex]
Substitute these values into the vertex formula:
[tex]\[ t = -\frac{0.5136}{2 \times -0.024} \][/tex]
Solving this, we find:
[tex]\[ t = 10.7 \][/tex]
So, the patient's temperature reaches its maximum value 10.7 hours after the illness begins.
### Finding the Maximum Temperature
To find the maximum temperature, substitute [tex]\( t = 10.7 \)[/tex] back into the temperature function [tex]\( T(t) \)[/tex]:
[tex]\[ T(10.7) = -0.024 \times (10.7)^2 + 0.5136 \times 10.7 + 97.9 \][/tex]
Calculate this expression to find:
[tex]\[ T(10.7) = 100.6 \][/tex]
Therefore, the patient's maximum temperature during the illness is 100.6 degrees Fahrenheit.
In summary:
- The patient's temperature reaches its maximum 10.7 hours after the illness begins.
- The maximum temperature is 100.6°F.
