Find theta prime of (282)

Question

A building that is 225 feet tall casts a shadow of various lengths as the day goes by. An angle of elevation  is formed by lines from the top and bottom of the building to the tip of the shadow, as seen in the figure above.

Find the rate of change of the angle of elevationwhen 282.

282 radians per foot. Round to five decimal places.

Answer 1

tanθ =225/x; sec²θdθ/dx = - 225/x²; dθ/dx = - 225cos²θ/x²; dθ/dx = - [225x²/(x² + 225²)]/x².

dθ/dx = - 225/(x² + 225²) = - 225/(225² + 282²) = - 0.001729 rad/foot