Case Study - 3 During a heavy gaming session, the temperature of a student's laptop processor increases significantly. After the session, the processor begins to cool down, and the rate of cooling is proportional to the difference between the processor's temperature and the room temperature ($25^\circ C$). Initially the processor's temperature is $85^\circ C$. The rate of cooling is defined by the equation $\frac{d}{dt}(T(t)) = -k(T(t) - 25)$, where $T(t)$ represents the temperature of the processor at time $t$ (in minutes) and $k$ is a constant. Based on the above information, answer the following questions : (i) Find the expression for temperature of processor, $T(t)$, given that $T(0) = 85^\circ C$. (ii) How long will it take for the processor's temperature to reach $40^\circ C$ ? Given that $k = 0.03$, $\log_e 4 = 1.3863$.

Case Study - 3 During a heavy gaming session, the tempe — CBSE Class 12 Mathematics 2025 PYQ

Question

25^\circ C

(i) Find the expression for temperature of processor,

T(t)

, given that

T(0) = 85^\circ C

(ii) How long will it take for the processor's temperature to reach

40^\circ C

? Given that

k = 0.03

\log_e 4 = 1.3863

CBSECBSE Class XII Board 2025Subjective· 4mImportance★★★★★

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Differential Equation Modeling

Differential Equation Modeling: From Intuition to Precision

Imagine you're watching a cup of hot coffee cool down on your desk. It starts hot, then gradually loses heat to the room. But here's the key question: how fast does it cool at any given moment? The answer isn't a fixed number — it depends on how hot the coffee is right now. The hotter it is, the faster it cools. This is the core idea behind differential equation modeling: the rate of change of a quantity depends on the quantity itself.

The Intuition First

Suppose the room is at $20^\circ$ C and your coffee starts at $90^\circ$ C. Newton's Law of Cooling says the rate at which the coffee cools is proportional to the temperature difference between the coffee and the room. So when the coffee is $90^\circ$ C, the difference is $70^\circ$ C and it cools fast; when it's $40^\circ$ C, the difference is only $20^\circ$ C and it cools slowly. The rate changes as the temperature changes.

If we let $T(t)$ be the temperature at time $t$ , then "rate of change of temperature" is $\frac{dT}{dt}$ , and the statement above becomes:

\frac{dT}{dt} = -k(T - 20)

The minus sign is because the temperature is decreasing; the constant $k$ depends on the cup, the liquid, etc. This is a differential equation — an equation that involves a function and its derivative.

The Precise Statement

A differential equation is any equation that contains an unknown function and one or more of its derivatives. The goal is to find the function itself (here, $T(t)$ ) that satisfies the equation.

\frac{dy}{dx} = f(x, y)

Modeling means taking a real-world situation and translating it into a differential equation:

Identify the quantity you want to study (population, temperature, drug concentration, voltage).
State the rate law in words: "The population grows at a rate proportional to its size."
Translate into math: $\frac{dP}{dt} = kP$ .
Add initial conditions: $P(0) = P_0$ (the starting value).

Why This Matters

Without differential equations, you'd have to guess the whole future behavior of a system. With them, you get a precise rule that governs every instant of change. The solution to $\frac{dP}{dt} = kP$ is $P(t) = P_0 e^{kt}$ — exponential growth. The solution to the cooling equation is $T(t) = 20 + 70 e^{-kt}$ — exponential decay toward room temperature.

Note

Most real-world models are not this simple. But every model starts the same way: observe how something changes, express that change in terms of the thing itself, and write it as a differential equation.

A Simple Example to Try

A bacteria culture doubles every hour. If you start with 100 bacteria:

Step 1: Quantity = population $P(t)$ . …

Start your 14-day free trial to unlock the full solution →

This is Newton’s Law of Cooling. The temperature decays exponentially from $85^\circ C$ toward the room temperature $25^\circ C$ . The solution is $T(t) = 25 + 60 e^{-kt}$ , and with $k = 0.03$ , it takes about $46.21$ minutes to reach $40^\circ C$ .

The core idea here is Newton’s Law of Cooling: the rate of change of an object’s temperature is proportional to the difference between its temperature and the surrounding environment. That’s exactly what the given differential equation says:

\frac{dT}{dt} = -k (T - 25)

The minus sign tells us the temperature decreases when $T > 25$ (which it is, initially). The constant $k > 0$ controls how fast cooling happens.

This is a first-order linear differential equation — and more specifically, it’s separable. That means we can rearrange it so all the $T$ terms are on one side and all the $t$ terms on the other, then integrate.

1. Solve the differential equation

Separate variables:

\frac{dT}{T - 25} = -k \, dt

Integrate both sides:

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

The left side gives $\ln |T - 25|$ , and the right side gives $-kt + C$ :

\ln |T - 25| = -kt + C

Since $T > 25$ throughout (the processor starts at $85^\circ C$ and cools toward $25^\circ C$ ), we can drop the absolute value:

\ln (T - 25) = -kt + C

Exponentiate both sides:

T - 25 = e^{-kt + C} = e^C e^{-kt}

Let $A = e^C$ , a positive constant:

T(t) = 25 + A e^{-kt}

2. Use the initial condition to find $A$

We know $T(0) = 85$ :

85 = 25 + A e^{0} \quad\Rightarrow\quad 85 = 25 + A \quad\Rightarrow\quad A = 60

So the temperature function is:

\boxed{T(t) = 25 + 60 e^{-kt}}

T(t) = T_{\text{room}} + (T_0 - T_{\text{room}}) e^{-kt}

This is the standard Newton’s Law of Cooling formula — the difference from room temperature decays exponentially.

3. Find the time to reach $40^\circ C$

We are given $k = 0.03$ . Set $T(t) = 40$ :

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

Subtract 25:

15 = 60 e^{-0.03 t}

Divide by 60:

Start your 14-day free trial to unlock the full solution →

1. Solve the differential equation

2. Use the initial condition to find AAA

3. Find the time to reach 40∘C40^\circ C40∘C

🎓Unlock everything free for 14 days

2. Use the initial condition to find $A$

3. Find the time to reach $40^\circ C$

Unlock everything free for 14 days