Question

A building that is 245 feet tall casts a shadow of various lengths x as the day goes by. An angle of elevation θ is formed by lines from the top and bottom of the building to the tip of the shadow. Find the rate of change (in radians per foot) of the angle of elevation dθ/dx when x = 300 feet.

Answer 1

Answer:

$\dfrac{d\theta}{dx} = -0.001633\ rad/s$

Step-by-step explanation:

given,

height of the building = 245 ft

distance between the shadow and the bottom of the building = x

to find angle of elevation dθ/dx when x = 300 feet.

$tan \theta = \dfrac{P}{B}$

$tan \theta = \dfrac{245}{x}$

$\theta = tan^{-1}(\dfrac{245}{x})$

$\dfrac{d}{dx}(tan^{-1} x) = \dfrac{1}{1 + x^2}$

$\dfrac{d\theta}{dx} = \dfrac{1}{1 +(\dfrac{245}{x})^2}\times \dfrac{-245}{x^2}$

$\dfrac{d\theta}{dx} = \dfrac{-245}{x^2+60025}$

$\dfrac{d\theta}{dx} = -0.001633\ rad/s$