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Answer :
To find when the patient's temperature reaches its maximum value and what that maximum temperature is, we can analyze the temperature function given:
[tex]\[ T(t) = -0.019t^2 + 0.4218t + 97.3 \][/tex]
The function is a quadratic equation in the form of [tex]\( ax^2 + bx + c \)[/tex]. Since the coefficient of the [tex]\( t^2 \)[/tex] term ([tex]\( a = -0.019 \)[/tex]) is negative, the parabola opens downwards, and therefore the vertex of the parabola represents the maximum point.
### Step-by-step Solution:
1. Determine the Time of Maximum Temperature:
For a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], the time [tex]\( t \)[/tex] at which the maximum value occurs can be found using the formula for the vertex:
[tex]\[ t = -\frac{b}{2a} \][/tex]
- Here, [tex]\( a = -0.019 \)[/tex] and [tex]\( b = 0.4218 \)[/tex].
Calculate [tex]\( t \)[/tex]:
[tex]\[ t = -\frac{0.4218}{2 \times -0.019} \][/tex]
[tex]\[ t \approx 11.1 \][/tex]
So, the patient's temperature reaches its maximum after approximately 11.1 hours.
2. Calculate the Maximum Temperature:
Now, substitute [tex]\( t = 11.1 \)[/tex] back into the temperature function to find the maximum temperature:
[tex]\[ T(11.1) = -0.019 \times (11.1)^2 + 0.4218 \times 11.1 + 97.3 \][/tex]
[tex]\[ T(11.1) \approx 99.6 \][/tex]
Therefore, the maximum temperature during the illness is approximately 99.6 degrees Fahrenheit.
### Answers:
- The patient's temperature reaches its maximum after 11.1 hours.
- The patient's maximum temperature is 99.6 degrees Fahrenheit.
[tex]\[ T(t) = -0.019t^2 + 0.4218t + 97.3 \][/tex]
The function is a quadratic equation in the form of [tex]\( ax^2 + bx + c \)[/tex]. Since the coefficient of the [tex]\( t^2 \)[/tex] term ([tex]\( a = -0.019 \)[/tex]) is negative, the parabola opens downwards, and therefore the vertex of the parabola represents the maximum point.
### Step-by-step Solution:
1. Determine the Time of Maximum Temperature:
For a quadratic equation in the form [tex]\( ax^2 + bx + c \)[/tex], the time [tex]\( t \)[/tex] at which the maximum value occurs can be found using the formula for the vertex:
[tex]\[ t = -\frac{b}{2a} \][/tex]
- Here, [tex]\( a = -0.019 \)[/tex] and [tex]\( b = 0.4218 \)[/tex].
Calculate [tex]\( t \)[/tex]:
[tex]\[ t = -\frac{0.4218}{2 \times -0.019} \][/tex]
[tex]\[ t \approx 11.1 \][/tex]
So, the patient's temperature reaches its maximum after approximately 11.1 hours.
2. Calculate the Maximum Temperature:
Now, substitute [tex]\( t = 11.1 \)[/tex] back into the temperature function to find the maximum temperature:
[tex]\[ T(11.1) = -0.019 \times (11.1)^2 + 0.4218 \times 11.1 + 97.3 \][/tex]
[tex]\[ T(11.1) \approx 99.6 \][/tex]
Therefore, the maximum temperature during the illness is approximately 99.6 degrees Fahrenheit.
### Answers:
- The patient's temperature reaches its maximum after 11.1 hours.
- The patient's maximum temperature is 99.6 degrees Fahrenheit.
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