Step 1: Separate Variables

We start with the given differential equation:

$\frac{d\theta}{dt} = \lambda(120 - \theta)$

We can rearrange this to separate variables:

$\frac{d\theta}{120 - \theta} = \lambda dt$

Step 2: Integrate Both Sides

Integrate both sides:

$\int \frac{d\theta}{120 - \theta} = \int \lambda dt$

This results in:

$-\ln(120 - \theta) = \lambda t + C$

Step 3: Solve for θ

By exponentiating both sides, we obtain:

$120 - \theta = e^{-\lambda t - C}$

Letting $A = e^{-C}$, we rearrange it to:

$\theta = 120 - Ae^{-\lambda t}$

Step 4: Apply Initial Condition

Using the initial condition where (\theta = 20) when (t = 0):

$20 = 120 - A \Rightarrow A = 100$

Thus, we can rewrite our equation as:

$\theta = 120 - 100e^{-\lambda t}$

This shows the required result.