ANSWERS AND EXPLANATIONS TO SECTION II
- The temperature on New Year’s Day in Hinterland was given by T(H) = −A − B cos ,
where T is the temperature in degrees Fahrenheit and H is the number of hours from midnight
(0 ≤ H < 24).
(a) The initial temperature at midnight was −15° F, and at noon of New Year’s Day was 5° F.
Find A and B.
Simply plug in the temperature, −15, for T and the time, midnight (H = 0), for H into the
equation. We get −15 = −A − B cos 0, which simplifies to −15 = −A − B.
Now plug the temperature, 5, for T and the time, noon (H = 12), for H into the equation. We get
5 = −A − B cos(π), which simplifies to 5 = −A + B.
Now we can solve the pair of simultaneous equations for A and B, and we get A = 5° F and B =
10° F.
(b) Find the average temperature for the first 10 hours.
In order to find the average value, we use the Mean Value Theorem for Integrals, which says
that the average value of f(x) on the interval [a, b] is
f(x) dx
Here, we have dH.
Evaluating the integral, we get
(c) Use the Trapezoid Rule with 4 equal subdivisions to estimate T(H) dH.
The Trapezoid Rule enables us to approximate the area under a curve with a fair degree of
accuracy. The rule says that the area between the x-axis and the curve y = f(x) on the interval
