The equation given in the image is:

$T(t) = -0.014t^2 + 0.35t + 99$

This is a quadratic function, and its maximum value occurs at the vertex because the coefficient of $t^2$ is negative, indicating that the parabola opens downward.

To find the time $t$ when the patient's temperature reaches its maximum, we can use the formula for the vertex of a quadratic function, which is given by:

$t = \frac{-b}{2a}$

For the function $T(t) = -0.014t^2 + 0.35t + 99$, the coefficients are:

• $a = -0.014$
• $b = 0.35$

Now, let's calculate the time $t$ when the maximum temperature occurs.The patient's temperature reaches its maximum value approximately 12.5 hours after the illness begins.

Next, to find the maximum temperature during the illness, we substitute this value of $t = 12.5$ into the original function $T(t)$. Let's calculate that.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to guide you through solving the maximum temperature manually, or is there anything else you'd like to ask?

Here are some related questions to expand on this topic:

1. How can you confirm the maximum using a graphing calculator?
2. What role does the coefficient of $t^2$ play in determining the shape of the parabola?
3. How do you calculate the vertex of a quadratic function in general?
4. What other real-life applications involve quadratic functions?
5. How does the initial temperature $T(0)$ compare to the maximum temperature?

Tip: When solving quadratics, always check if the parabola opens upwards or downwards to determine if you're looking for a maximum or minimum!