A hot cup of coffee is placed in a room and cools according to the function \(\mathrm{T(t) = 70 +...

1. TRANSLATE the function structure

• Given: \(\mathrm{T(t) = 70 + 130e^{-0.05t}}\)
• This is an exponential decay function of the form \(\mathrm{a + be^{-kt}}\)
• Where \(\mathrm{a = 70, b = 130, k = 0.05}\)

2. INFER how to find the horizontal asymptote

• For exponential functions \(\mathrm{T(t) = a + be^{-kt}}\), the horizontal asymptote is always \(\mathrm{y = a}\)
• This happens because as t gets very large, \(\mathrm{e^{-kt}}\) approaches 0
• So \(\mathrm{T(t)}\) approaches \(\mathrm{a + b(0) = a}\)

3. Calculate the horizontal asymptote

• From \(\mathrm{T(t) = 70 + 130e^{-0.05t}}\), we have \(\mathrm{a = 70}\)
• Therefore, the horizontal asymptote is \(\mathrm{y = 70}\)

4. INFER the contextual meaning

• This is a coffee cooling problem - the coffee starts hot and cools down
• The horizontal asymptote represents the temperature the coffee will eventually reach
• In cooling problems, objects approach the temperature of their surroundings
• So \(\mathrm{y = 70}\) represents the room temperature

Answer: D - The estimated room temperature is 70 degrees.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students correctly identify the horizontal asymptote as \(\mathrm{y = 70}\), but misinterpret what this represents in context. They might think it represents the initial temperature of the coffee rather than the final equilibrium temperature.

Students see that when t = 0,

\(\mathrm{T(0) = 70 + 130e^0}\)

\(\mathrm{= 70 + 130}\)

\(\mathrm{= 200°F}\)

which is the initial coffee temperature. They then confuse the horizontal asymptote value (70) with this initial condition, leading them to select Choice C (The estimated temperature of the coffee when it was first placed in the room was 200 degrees) - which happens to be numerically correct but interprets the wrong part of the function.

Second Most Common Error:

Incomplete TRANSLATE reasoning: Students may identify that 200°F is the initial temperature and incorrectly think the horizontal asymptote should represent this maximum temperature rather than the limiting temperature.

This leads them to select Choice A (The maximum estimated temperature of the coffee during the 60 minutes was 200 degrees) because they recognize 200 as significant but miss that horizontal asymptotes describe long-term behavior, not maximums.

The Bottom Line:

This problem tests whether students understand that horizontal asymptotes in decay contexts represent equilibrium values (what the quantity approaches), not initial conditions or maximums. The key insight is connecting the mathematical limit to the physical reality of thermal equilibrium.