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] hours after the illness begins is given by the function:

[tex]\[ T = -0.017t^2 + 0.425t + 97.3 \][/tex]

1. When does the patient's temperature, [tex]$T$[/tex], reach its maximum value?
- After [tex]$\_\_\_\_\_$[/tex] (Select an answer)

2. What is the patient's maximum temperature, [tex]$T$[/tex], during the illness?
- The maximum temperature is [tex]$\_\_\_\_\_$[/tex] (Select an answer)

We are given the quadratic function

[tex]$$
T(t) = -0.017t^2 + 0.425t + 97.3
$$[/tex]

which models the patient's temperature (in °F) [tex]$t$[/tex] hours after the illness begins.

Since the quadratic coefficient is negative ([tex]$a = -0.017$[/tex]), the parabola opens downward and the maximum temperature occurs at the vertex of the parabola.

For a quadratic function of the form

[tex]$$
T(t) = at^2 + bt + c,
$$[/tex]

the vertex occurs at

[tex]$$
t = -\frac{b}{2a}.
$$[/tex]

Substitute the values [tex]$a = -0.017$[/tex] and [tex]$b = 0.425$[/tex]:

[tex]$$
t = -\frac{0.425}{2(-0.017)} = 12.5 \text{ hours}.
$$[/tex]

This means the patient's temperature reaches its maximum about [tex]$12.5$[/tex] hours after the illness begins.

To find the maximum temperature, substitute [tex]$t = 12.5$[/tex] back into the function:

[tex]$$
T(12.5) = -0.017(12.5)^2 + 0.425(12.5) + 97.3.
$$[/tex]

First, calculate [tex]$(12.5)^2$[/tex]:

[tex]$$
(12.5)^2 = 156.25.
$$[/tex]

Then compute each term:

1. First term:
[tex]$$
-0.017 \times 156.25 = -2.65625.
$$[/tex]

2. Second term:
[tex]$$
0.425 \times 12.5 = 5.3125.
$$[/tex]

Now, add these to the constant term:

[tex]$$
T(12.5) = -2.65625 + 5.3125 + 97.3 = 99.95625 \text{ °F}.
$$[/tex]

Thus, the maximum temperature is approximately [tex]$99.95625$[/tex] °F.

In summary:

- The patient's temperature reaches its maximum value at [tex]$t = 12.5$[/tex] hours.
- The maximum temperature is approximately [tex]$99.95625$[/tex] °F.