To solve this problem, let's analyze the given temperature function:
[tex]\[ T(t) = -0.012 t^2 + 0.288 t + 97.1 \][/tex]
This is a quadratic function of the form [tex]\( T(t) = at^2 + bt + c \)[/tex], where:
- [tex]\( a = -0.012 \)[/tex]
- [tex]\( b = 0.288 \)[/tex]
- [tex]\( c = 97.1 \)[/tex]
For a quadratic function [tex]\( ax^2 + bx + c \)[/tex], the maximum or minimum value (vertex) occurs at [tex]\( t = -\frac{b}{2a} \)[/tex]. Since the coefficient of [tex]\( t^2 \)[/tex] (i.e., [tex]\( a \)[/tex]) is negative, this quadratic function opens downwards and thus has a maximum value.
To find the time [tex]\( t \)[/tex] at which the maximum temperature occurs, we use:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the given values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ t = -\frac{0.288}{2(-0.012)} \][/tex]
Further simplifying, we get:
[tex]\[ t = -\frac{0.288}{-0.024} = 12.0 \][/tex]
This means the patient's temperature reaches its maximum value at [tex]\( t = 12.0 \)[/tex] hours after the illness begins.
Next, to find the maximum temperature, we substitute [tex]\( t = 12.0 \)[/tex] back into the original function:
[tex]\[ T(12.0) = -0.012 (12.0)^2 + 0.288 (12.0) + 97.1 \][/tex]
Calculating each term individually:
[tex]\[ -0.012 \times 144 + 0.288 \times 12 + 97.1 = -1.728 + 3.456 + 97.1 \][/tex]
Summing these values gives:
[tex]\[ -1.728 + 3.456 = 1.728 \][/tex]
[tex]\[ 1.728 + 97.1 = 98.8 \][/tex]
Thus, the maximum temperature the patient reaches during the illness is [tex]\( 98.8 \)[/tex] degrees Fahrenheit.
So, summarizing our answers:
1. The patient's temperature reaches its maximum value after [tex]\( 12.0 \)[/tex] hours.
2. The maximum temperature during the illness is [tex]\( 98.8 \)[/tex] degrees Fahrenheit.
