A hot beverage cools according to the law dtdT=−k(T−20), where T is temperature in degrees Celsius and k>0 is a constant. A slope field for this differential equation is shown for 0≤t≤60 minutes and 20≤T≤90.
• Horizontal arrows appear along the line T=20.
• Above T=20 the arrows tilt downward, becoming less steep as they approach T=20.
• No arrows point upward anywhere in the field.
(a) Explain why every integral curve in the field is decreasing for all t>0.
(b) Describe the long–term behavior of any solution that starts with T(0)=70. Include both the qualitative trend and the limiting value as t→∞, providing justification from the slope field.
(c) Suppose a second cup starts at T(0)=50. Use the slope field to compare the relative steepness of the two cooling curves during the first ten minutes, and explain what this indicates about their rates of cooling.