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Answer :
To determine when the patient's temperature reaches its maximum value and to find out what that maximum temperature is, we start by examining the given temperature function:
[tex]\[ T(t) = -0.024t^2 + 0.528t + 97.7 \][/tex]
This is a quadratic function that represents a parabola. Because the coefficient of [tex]\( t^2 \)[/tex] is negative, the parabola opens downward, indicating that it has a maximum point at its vertex.
### Finding the Time for Maximum Temperature:
For a quadratic equation in the form [tex]\( T(t) = at^2 + bt + c \)[/tex], the vertex (where the maximum or minimum temperature occurs) can be found using the formula for the t-coordinate of the vertex:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Given:
- [tex]\( a = -0.024 \)[/tex]
- [tex]\( b = 0.528 \)[/tex]
Plug these values into the formula to find [tex]\( t \)[/tex]:
[tex]\[ t = -\frac{0.528}{2 \times -0.024} \][/tex]
After calculating, we find that the time [tex]\( t \)[/tex] is approximately 11.0 hours.
### Finding the Maximum Temperature:
Now, substitute [tex]\( t = 11.0 \)[/tex] back into the original temperature function to find the maximum temperature:
[tex]\[ T(11.0) = -0.024(11.0)^2 + 0.528(11.0) + 97.7 \][/tex]
After performing these calculations, we find that the maximum temperature [tex]\( T \)[/tex] is approximately 100.6 degrees Fahrenheit.
### Conclusion:
- The patient's temperature reaches its maximum value after approximately 11.0 hours.
- The maximum temperature during the illness is approximately 100.6°F.
[tex]\[ T(t) = -0.024t^2 + 0.528t + 97.7 \][/tex]
This is a quadratic function that represents a parabola. Because the coefficient of [tex]\( t^2 \)[/tex] is negative, the parabola opens downward, indicating that it has a maximum point at its vertex.
### Finding the Time for Maximum Temperature:
For a quadratic equation in the form [tex]\( T(t) = at^2 + bt + c \)[/tex], the vertex (where the maximum or minimum temperature occurs) can be found using the formula for the t-coordinate of the vertex:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Given:
- [tex]\( a = -0.024 \)[/tex]
- [tex]\( b = 0.528 \)[/tex]
Plug these values into the formula to find [tex]\( t \)[/tex]:
[tex]\[ t = -\frac{0.528}{2 \times -0.024} \][/tex]
After calculating, we find that the time [tex]\( t \)[/tex] is approximately 11.0 hours.
### Finding the Maximum Temperature:
Now, substitute [tex]\( t = 11.0 \)[/tex] back into the original temperature function to find the maximum temperature:
[tex]\[ T(11.0) = -0.024(11.0)^2 + 0.528(11.0) + 97.7 \][/tex]
After performing these calculations, we find that the maximum temperature [tex]\( T \)[/tex] is approximately 100.6 degrees Fahrenheit.
### Conclusion:
- The patient's temperature reaches its maximum value after approximately 11.0 hours.
- The maximum temperature during the illness is approximately 100.6°F.
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