Final answer:
To find the average temperature, find the definite integral of the temperature function and divide it by the length of the interval. To find the minimum and maximum temperatures, find the vertex of the parabola.
Step-by-step explanation:
To find the average temperature, we need to find the average value of the function T(t) over the given 9-hour period. We can do this by finding the definite integral of T(t) from t = 0 to t = 9, and then dividing the result by the length of the interval (9 - 0 = 9). The average temperature is:
(1/9) * ∫[-t² + t + 31]dt from 0 to 9
Next, to find the minimum and maximum temperatures, we can find the vertices of the parabola represented by the function T(t). Since the coefficient of the t² term is negative, the parabola opens downwards and the vertex represents the maximum point. The minimum temperature is the temperature at t = 0 and the maximum temperature is the temperature at the x-coordinate of the vertex. The vertex of the parabola is:
x = -b/2a = -1/(2(-1)) = 0.5
The minimum temperature is T(0) = -0² + 0 + 31 = 31°C and the maximum temperature is T(0.5) = -(0.5)² + 0.5 + 31 = 31.125°C.