Question

37-40. {Use of Tech} Temperature dataHowdy temperature data for Boulder, Colorado; San Francisco, California; Nantucket, Massachusetts; and Duluth, Minnesota, over a 12-hr period on the same day of January are shown in the figure.Assume these data are taken from a continuous temperature function T(t). The average temperature (in °F) over the 12-hr period is:T_avg = (1/12) × ∫(0 to 12) T(t) dt38. Find an accurate approximation to the average temperature over the 12-hr period for San Francisco. State your method.

Accepted Answer

Identify the temperature data for San Francisco (SF) from the table, which gives temperature values at each hour from 0 to 12. Recall the formula for the average temperature over the 12-hour period: $$T_{avg} = \frac{1}{12} \$$int_0$$^{12} T(t) \, dt. $$Since the temperature data is discrete, approximate the integral $$\$$int_0$$^{12} T(t) \, dt$$ using a numerical integration method such as the Trapezoidal Rule or Simpson's Rule. For the Trapezoidal Rule, calculate the sum: $$\frac{\Delta t}{2} \left[ T(0) + 2T(1) + 2T(2) + \cdots + 2T(11) + T(12) \right]$$, where $$\Delta t = 1$$ hour. Finally, multiply the result of the numerical integration by $$\frac{1}{12} to $$find the average temperature $$T_{avg}$$ over the 12-hour period.

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Key Concepts

Definite Integral as Area Under a Curve

Average Value of a Function

Numerical Integration Methods (e.g., Trapezoidal Rule)