Answer
$\frac{d\theta}{dt} = 0.04$ rads/s
Work Step by Step
Step 1: Information
Let the height be $y$
$\frac{dy}{dt} = 4$
$tan(\theta) = \frac{y}{50} $
Step 2: Derive $tan(\theta) = \frac{y}{50} $
Through implicit differentiation with respect to $t$
$sec^2(\theta)\times\frac{d\theta}{dt} = \frac{1}{50} \times \frac{dy}{dt} $
Move the $sec^2(\theta)$ to the right-hand side
$\frac{d\theta}{dt} = cos^2(\theta) (\frac{1}{50} \times \frac{dy}{dt}) $
Step 3: Substitute values
$\frac{dy}{dt} = 4$
Find $\theta$
$\frac{d\theta}{dt} = cos^2(\theta) (\frac{2}{25})$
$\theta = tan^{-1}(\frac{50}{50})$
$\theta = tan^{-1}(1)$
$\theta = 0.785$
Substitute $\theta$
$\frac{d\theta}{dt} = cos^2(0.785) (\frac{2}{25})$
$\frac{d\theta}{dt} = 0.04$ rads/s
