Answer :
To solve the problem of finding when the patient's temperature reaches its maximum value and what that maximum temperature is, we need to analyze the given temperature function:
[tex]\[ T(t) = -0.019t^2 + 0.4712t + 98.3 \][/tex]
This function is a quadratic equation in the form of [tex]\( T(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -0.019 \)[/tex], [tex]\( b = 0.4712 \)[/tex], and [tex]\( c = 98.3 \)[/tex].
### Finding When the Temperature Reaches Its Maximum
Since the function is a downward-opening parabola (as indicated by the negative coefficient of [tex]\( t^2 \)[/tex]), it will have a maximum point at its vertex. The formula to find the vertex [tex]\( t \)[/tex] for the maximum or minimum value of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ t = -\frac{0.4712}{2 \times -0.019} \][/tex]
When calculated, this results in:
[tex]\[ t = 12.4 \][/tex]
This tells us that the patient's temperature reaches its maximum value after 12.4 hours.
### Finding the Maximum Temperature
Next, we substitute [tex]\( t = 12.4 \)[/tex] back into the function to find the maximum temperature:
[tex]\[ T(12.4) = -0.019(12.4)^2 + 0.4712(12.4) + 98.3 \][/tex]
After performing the calculations, the maximum temperature is found to be:
[tex]\[ T = 101.2 \][/tex] degrees Fahrenheit
### Conclusion
Thus, the patient's temperature reaches its maximum value after 12.4 hours, and the maximum temperature during the illness is 101.2 degrees Fahrenheit.
[tex]\[ T(t) = -0.019t^2 + 0.4712t + 98.3 \][/tex]
This function is a quadratic equation in the form of [tex]\( T(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -0.019 \)[/tex], [tex]\( b = 0.4712 \)[/tex], and [tex]\( c = 98.3 \)[/tex].
### Finding When the Temperature Reaches Its Maximum
Since the function is a downward-opening parabola (as indicated by the negative coefficient of [tex]\( t^2 \)[/tex]), it will have a maximum point at its vertex. The formula to find the vertex [tex]\( t \)[/tex] for the maximum or minimum value of a quadratic equation [tex]\( ax^2 + bx + c \)[/tex] is:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Plugging the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ t = -\frac{0.4712}{2 \times -0.019} \][/tex]
When calculated, this results in:
[tex]\[ t = 12.4 \][/tex]
This tells us that the patient's temperature reaches its maximum value after 12.4 hours.
### Finding the Maximum Temperature
Next, we substitute [tex]\( t = 12.4 \)[/tex] back into the function to find the maximum temperature:
[tex]\[ T(12.4) = -0.019(12.4)^2 + 0.4712(12.4) + 98.3 \][/tex]
After performing the calculations, the maximum temperature is found to be:
[tex]\[ T = 101.2 \][/tex] degrees Fahrenheit
### Conclusion
Thus, the patient's temperature reaches its maximum value after 12.4 hours, and the maximum temperature during the illness is 101.2 degrees Fahrenheit.
