Answer :
To solve this problem and find when the patient's temperature reaches its maximum value and what that maximum temperature is, we need to analyze the function provided:
[tex]\[ T(t) = -0.015t^2 + 0.315t + 98.2 \][/tex]
This is a quadratic function, which is typically a parabola. Because the coefficient of [tex]\(t^2\)[/tex] is negative, the parabola opens downwards, meaning it has a maximum point.
### Step 1: Finding the time when the maximum temperature occurs
For any quadratic function of the form [tex]\(ax^2 + bx + c\)[/tex], the maximum or minimum value occurs at:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In this case:
- [tex]\(a = -0.015\)[/tex]
- [tex]\(b = 0.315\)[/tex]
Let's plug these values into the formula:
[tex]\[ t = -\frac{0.315}{2 \times (-0.015)} \][/tex]
After evaluating this expression, we find:
[tex]\[ t \approx 10.5 \][/tex]
Thus, the patient's temperature reaches its maximum value 10.5 hours after the illness begins.
### Step 2: Finding the maximum temperature
Now we use the time [tex]\(t = 10.5\)[/tex] to calculate the maximum temperature by substituting it back into the original temperature function:
[tex]\[ T(10.5) = -0.015(10.5)^2 + 0.315(10.5) + 98.2 \][/tex]
By calculating this expression, we get:
[tex]\[ T(10.5) \approx 99.9 \][/tex]
So, the patient's maximum temperature during the illness is approximately 99.9 degrees Fahrenheit.
In summary, the patient's temperature reaches its maximum value after 10.5 hours, and the maximum temperature is 99.9 degrees Fahrenheit.
[tex]\[ T(t) = -0.015t^2 + 0.315t + 98.2 \][/tex]
This is a quadratic function, which is typically a parabola. Because the coefficient of [tex]\(t^2\)[/tex] is negative, the parabola opens downwards, meaning it has a maximum point.
### Step 1: Finding the time when the maximum temperature occurs
For any quadratic function of the form [tex]\(ax^2 + bx + c\)[/tex], the maximum or minimum value occurs at:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In this case:
- [tex]\(a = -0.015\)[/tex]
- [tex]\(b = 0.315\)[/tex]
Let's plug these values into the formula:
[tex]\[ t = -\frac{0.315}{2 \times (-0.015)} \][/tex]
After evaluating this expression, we find:
[tex]\[ t \approx 10.5 \][/tex]
Thus, the patient's temperature reaches its maximum value 10.5 hours after the illness begins.
### Step 2: Finding the maximum temperature
Now we use the time [tex]\(t = 10.5\)[/tex] to calculate the maximum temperature by substituting it back into the original temperature function:
[tex]\[ T(10.5) = -0.015(10.5)^2 + 0.315(10.5) + 98.2 \][/tex]
By calculating this expression, we get:
[tex]\[ T(10.5) \approx 99.9 \][/tex]
So, the patient's maximum temperature during the illness is approximately 99.9 degrees Fahrenheit.
In summary, the patient's temperature reaches its maximum value after 10.5 hours, and the maximum temperature is 99.9 degrees Fahrenheit.
