A city’s hourly temperature (in degrees Celsius) is modeled by T(t)=18+7 ! ( t 12 )
MATHEMATICS · Multi-part · A Level · Mathematics · Exponentials and logarithms · Exponentials and logarithms
Problem
A city’s hourly temperature (in degrees Celsius) is modeled by
T(t)=18+7sin(12πt),
where t is the number of hours after noon. Negative values of t correspond to times before noon, so the model is valid for −12≤t≤12.
(a) Identify which part of T(t) is responsible for fluctuations about a baseline value. Explain how the symmetry of this fluctuating part influences the integral of T over the full 24-hour period represented by t∈[−12,12].?
(b) Determine the average temperature for this 24-hour period. Show all work leading to your result.
(c) The city issues an advisory when the temperature exceeds the daily average. Solve the inequality T(t)>(average temperature) for t∈[−12,12], and state the total number of hours (to the nearest tenth) during which the advisory is in effect.