The quadratic model of temperatures is T(t)=-0.008x^2+0.237x+56.97. Highest temperature at 2:20 PM.
a) Quadratic model for the data
We can use the following quadratic model to fit the data:
T(t) = a * t^2 + b * t + c
where T(t) is the predicted temperature at time t, and a, b, and c are the model parameters.
To fit the model to the data, we can use the following least squares method:
1. Calculate the sum of squares of the residuals, which is the sum of the squares of the differences between the predicted temperatures and the actual temperatures:
SSR = sum((T(t_i) - T_i)^2)
where T_i is the actual temperature at time t_i.
2. Minimize the SSR by adjusting the model parameters a, b, and c.
Using the least squares method, we obtain the following model parameters:
a = -0.00833
b = 0.2375
c = 56.96667
Therefore, the quadratic model for the data is:
T(t) = -0.00833 * t^2 + 0.2375 * t + 56.96667
b) Time when the temperature is the highest
The temperature is highest when the derivative of the quadratic model is equal to zero. Therefore, we need to solve the following equation:
T'(t) = 0
where T'(t) is the derivative of T(t).
Solving the equation, we obtain t = 14.28. This means that the temperature is highest at approximately 2:20 PM.