Thank you for visiting A patient has an illness that typically lasts about 24 hours The temperature tex T tex in degrees Fahrenheit of the patient tex t tex. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
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.
Thank you for reading the article A patient has an illness that typically lasts about 24 hours The temperature tex T tex in degrees Fahrenheit of the patient tex t tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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