Meteorologists model the air temperature (in degrees Celsius) at a certain coastal…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Meteorologists model the air temperature (in degrees Celsius) at a certain coastal location by the function
T(x)=10+5cos(12πx),
where x is the number of hours after midnight (so 0≤x≤24 corresponds to one full day).
They wish to identify a consecutive block of morning hours, beginning at midnight (x=0), during which the average temperature equals 12∘C.
Parts:
(a) Write an integral expression for the average temperature, in degrees Celsius, experienced from x=0 to x=L hours after midnight.
(b) Evaluate the integral in part (a) and simplify the resulting average temperature expression so that it depends only on L.
(c) Determine the value of L (with 0<L<12) that yields an average temperature of 12∘C. Clearly justify all steps leading to your result.
(d) State the specific time interval (in clock time) during which this average temperature condition is met, and discuss whether more than one such interval could satisfy the requirement within a single 24-hour period.