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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:

[tex]T(t) = -0.022t^2 + 0.528t + 97.7[/tex]

Find the vertex of the parabola described by this function.

Answer :

Final answer:

The vertex of the given quadratic equation, which represents the peak temperature of a patient's illness, occurs roughly 12 hours into the illness and reaches a temperature of around 98.1°F.

Explanation:

The equation given represents a parabola, and its vertex can be found using the formula: h=-b/2a , where a and b are coefficients of the quadratic equation (in this case, a = -0.022, b = 0.528). Thus, the t-coordinate of the vertex (h) is -(0.528) / 2*(-0.022) which results in approximately 12 hours. To find the temperature at the vertex (k), substitute h into the equation: T(12) = -0.022*12^2 + 0.528*12 + 97.7 which gives about 98.1°F. Hence, the vertex of the parabola is (12, 98.1), meaning the patient's temperature will be at its peak 12 hours into the illness and will reach approximately 98.1°F.

Learn more about Vertex of a Parabola here:

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