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: \[ T(t)=-0.017 t^{2}+0.4182 t+97.2 \] Round all answers to 1 decimal place. When does the patient's temperature reach it maximum value? Answer: After Select an answerv What is the patient's maximum temperature during the illness? Answer: Select an answerv

Step 1 ：The temperature function is a quadratic function in the form of \(f(x) = ax^2 + bx + c\). The maximum value of a quadratic function is at its vertex. The x-coordinate of the vertex of a quadratic function is given by \(-b/2a\). In this case, \(a = -0.017\) and \(b = 0.4182\). So, we can calculate the time when the patient's temperature reaches its maximum value by using the formula \(-b/2a\).

Step 2 ：Substitute \(a = -0.017\) and \(b = 0.4182\) into the formula \(-b/2a\) to get the time when the patient's temperature reaches its maximum value.

Step 3 ：The result is approximately 12.3 hours. This means that the patient's temperature reaches its maximum value about 12.3 hours after the illness begins.

Step 4 ：Final Answer: The patient's temperature reaches its maximum value after \(\boxed{12.3}\) hours.