On a spring afternoon the interior temperature of a parked car is modeled by a…
MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability
Problem
On a spring afternoon the interior temperature of a parked car is modeled by a continuous function T(t), in °F, for 0≤t≤4, where t is hours after 1:00 p.m. The temperature readings at the endpoints of the interval are T(0)=68 and T(4)=102.
(a) Explain why the car’s interior temperature was exactly 90 °F at least once between 1:00 p.m. and 5:00 p.m. Provide detailed reasoning.
(b) Show that there was at least one time when the temperature was 80 °F or lower and a later time when it was 80 °F or higher. Then conclude that at some time the temperature was precisely 80 °F. Justify each step.
(c) Passengers may safely enter the car only when the interior temperature is between 70 °F and 85 °F inclusive. Based on the information provided, discuss whether it is guaranteed that such a safe time occurred during the four-hour interval and support your conclusion.