We appreciate your visit to 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 offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To find when the patient's temperature reaches its maximum value and what that maximum temperature is, we'll work with the given quadratic function:
[tex]\[ T(t) = -0.021t^2 + 0.5124t + 98.3 \][/tex]
### Step 1: Determine when the maximum temperature occurs
Since the function is quadratic in the form [tex]\( ax^2 + bx + c \)[/tex], the maximum or minimum value of a quadratic function occurs at:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -0.021 \)[/tex] and [tex]\( b = 0.5124 \)[/tex]. Plug these values into the formula:
[tex]\[ t = -\frac{0.5124}{2 \times -0.021} \][/tex]
After calculating, we find that:
[tex]\[ t \approx 12.2 \][/tex]
So, the patient’s temperature reaches its maximum value approximately 12.2 hours after the illness begins.
### Step 2: Find the maximum temperature
To find the maximum temperature, substitute [tex]\( t = 12.2 \)[/tex] back into the temperature function:
[tex]\[ T(12.2) = -0.021(12.2)^2 + 0.5124 \times 12.2 + 98.3 \][/tex]
Calculate the result:
[tex]\[ T(12.2) \approx 101.4 \][/tex]
Therefore, the patient's maximum temperature during the illness is approximately 101.4 degrees Fahrenheit.
### Summary
- The patient's temperature reaches the maximum approximately 12.2 hours after the illness begins.
- The maximum temperature during the illness is approximately 101.4 degrees Fahrenheit.
[tex]\[ T(t) = -0.021t^2 + 0.5124t + 98.3 \][/tex]
### Step 1: Determine when the maximum temperature occurs
Since the function is quadratic in the form [tex]\( ax^2 + bx + c \)[/tex], the maximum or minimum value of a quadratic function occurs at:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Here, [tex]\( a = -0.021 \)[/tex] and [tex]\( b = 0.5124 \)[/tex]. Plug these values into the formula:
[tex]\[ t = -\frac{0.5124}{2 \times -0.021} \][/tex]
After calculating, we find that:
[tex]\[ t \approx 12.2 \][/tex]
So, the patient’s temperature reaches its maximum value approximately 12.2 hours after the illness begins.
### Step 2: Find the maximum temperature
To find the maximum temperature, substitute [tex]\( t = 12.2 \)[/tex] back into the temperature function:
[tex]\[ T(12.2) = -0.021(12.2)^2 + 0.5124 \times 12.2 + 98.3 \][/tex]
Calculate the result:
[tex]\[ T(12.2) \approx 101.4 \][/tex]
Therefore, the patient's maximum temperature during the illness is approximately 101.4 degrees Fahrenheit.
### Summary
- The patient's temperature reaches the maximum approximately 12.2 hours after the illness begins.
- The maximum temperature during the illness is approximately 101.4 degrees Fahrenheit.
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