How long will it take until our measurements of the coffee’s temperature are indistinguishable from room temperature?

Background: When a function f(t) represents some real-world quantity, its limit as 𝑡→∞t→∞ represents the “long-term” behavior. Often, this kind of limit can be evaluated through algebraic methods. However, in more difficult cases, limits can be evaluated in terms of their individual parts by applying limit laws (see p. 95 from Section 2.3).

As we now know, limits involving infinity can be evaluated with analytic techniques (see Section 2.6), without the need for graphs or numerical calculations. By combining these methods, we can analyze some interesting physical problems.

The Application: A cup of coffee is brought into a room with constant temperature 20∘C20∘C. The temperature of the coffee after t minutes is given by

𝑇(𝑡)=20+60𝑒𝑡/5.T(t)=20+60et/5.

Answer each of the questions about this scenario, being sure to show all your work. Use exact values whenever possible, though you may give final answers in decimal form (use at least 3 places after the decimal point).

Your Task. Your solution needs to include all of the following steps. For each one, show all the math that is required in order to get your answer.

Limit Laws. Use the limit laws from Section 2.3 to evaluate lim𝑡→∞𝑇(𝑡)limt→∞T(t), being sure to justify each step with a limit law (see pages 95-96). Hint: you will need to evaluate the limit lim𝑡→∞𝑒−𝑡limt→∞e−t at some point — use what you know from Pre-Calculus to get the answer here.

Interpretation. In 2-3 complete sentences, explain the physical meaning of the calculation you did in Step 1. What happens to the coffee in this scenario? Why?

Equation-Solving. Suppose that our thermometer has a minimum sensitivity of 0.05∘C0.05∘C — in other words, it can’t detect a difference in temperature that is less than this amount. How long will it take until our measurements of the coffee’s temperature are indistinguishable from room temperature?