Questions: A patient has an illness that typically lasts about 24 hours. The temperature, T, in degrees Fahrenheit, of the patient t hours after the illness begins is given by: T(t)=-0.011 t^2+0.2376 t+98.6. Use your calculator to graph the function and answer the following questions. Round all answers to 1 decimal place. When does the patient's temperature reach its maximum value? Answer: After What is the patient's maximum temperature during the illness? Answer:

Solution Steps

To find the time when the temperature reaches its maximum value, we need to find the vertex of the parabola given by the function \( T(t) = -0.011 t^2 + 0.2376 t + 98.6 \). The vertex form of a parabola \( at^2 + bt + c \) has its maximum or minimum at \( t = -\frac{b}{2a} \).

Given: \[ a = -0.011 \] \[ b = 0.2376 \]

The time \( t \) at which the temperature is maximum is: \[ t = -\frac{0.2376}{2 \times -0.011} = \frac{0.2376}{0.022} \approx 10.8 \]

To find the maximum temperature, we substitute \( t = 10.8 \) back into the function \( T(t) \).

\[ T(10.8) = -0.011(10.8)^2 + 0.2376(10.8) + 98.6 \] \[ T(10.8) = -0.011(116.64) + 2.56528 + 98.6 \] \[ T(10.8) = -1.2830 + 2.5653 + 98.6 \] \[ T(10.8) \approx 99.9 \]

Final Answer