Step-by-Step Solution

Topic: Separation of Variables & Cooling Models

(I) DERIVING THE EXPRESSION

1. Separate and Integrate:

\[ \int \frac{dT}{T – 25} = \int -k \, dt \]

\( \ln|T – 25| = -kt + C_1 \)

2. Solve for T:

\( T – 25 = C \cdot e^{-kt} \)

3. Use Initial Condition \( T(0) = 85 \):

\( 85 = 25 + C \cdot e^0 \)   ⇒   C = 60

(II) CALCULATING TIME TO REACH 40°C

Substitute \( T(t) = 40 \) and \( k = 0.03 \):

\( 40 = 25 + 60e^{-0.03t} \)

\( 15 = 60e^{-0.03t} \)   ⇒   \( \frac{1}{4} = e^{-0.03t} \)

Take Natural Log (ln) of both sides:

\( \ln(1/4) = -0.03t \)   ⇒   \( -\ln(4) = -0.03t \)

\( -1.3863 = -0.03t \)

⚠️ Silly Mistake Audit: The “Ambient” Error

Always ensure your final temperature approaches the room temperature (25°C) as time goes to infinity. If your formula doesn’t have the “+25” term, your laptop would eventually reach 0°C, which is impossible!