The temperature at a chemical reactor is monitored in degrees Celsius for 10 minutes…
MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
The temperature at a chemical reactor is monitored in degrees Celsius for 10 minutes during a critical phase. Engineers later convert the recording to degrees Fahrenheit using F(t)=59C(t)+32. They wish to be certain that the converted temperature profile is continuous so that subsequent analysis using derivatives is legitimate.
(a) Suppose the Celsius recording is modeled by
C(t)={140−3t,110+5sin(6π(t−4)),0≤t≤4,4<t≤10,
where t is measured in minutes and C in °C. Confirm that C(t) is continuous on [0,10].
(b) Using the formula relating F and C, construct the explicit piecewise formula for F(t).
(c) Prove that F(t) is continuous on [0,10] and explain why continuity of C(t) is sufficient to guarantee continuity of F(t). Reference specific properties of the conversion function in your argument.
(d) Knowing F(0)=284 °F and F(10)=230 °F, demonstrate that there exists a time during the 10-minute interval when the Fahrenheit temperature was exactly 250 °F. State any theorem you invoke and clearly articulate the required hypotheses.