Answer:
To solve this problem, we can use trigonometry and differentiation. Let θ be the angle of elevation formed by the lines from the top and bottom of the building to the tip of the shadow. Then, we have:
tan(θ) = height of building / length of shadow
Differentiating both sides with respect to x, we get:
sec^2(θ) dθ/dx = (-1 / length of shadow^2) (d length of shadow / dx) height of building
Substituting the given values, we get:
sec^2(θ) dθ/dx = (-1 / x^2) (d x / dx) 235
At x = 286, we have:
length of shadow = x + 235 tan(θ)
Differentiating this expression with respect to x, we get:
d length of shadow / dx = 1 + 235 sec^2(θ) dθ/dx
Substituting this into the previous equation and simplifying, we get:
dθ/dx = - x^2 / (235 (x + 235 tan(θ)))
At x = 286, we have:
length of shadow = 286 + 235 tan(θ)
tan(θ) = height of building / length of shadow = 235 / (286 + 235 tan(θ))
Solving for tan(θ), we get:
tan(θ) = 235 / (286 + 235 tan(θ))
tan(θ) (286 + 235 tan(θ)) = 235
235 tan^2(θ) + 286 tan(θ) - 235 = 0
Using the quadratic formula, we get:
tan(θ) = 0.470835 or -1.00084
Since the angle of elevation is positive, we take:
tan(θ) = 0.470835
Substituting this into the expression for dθ/dx, we get:
dθ/dx = - 286^2 / (235 (286 + 235 (0.470835)))
Simplifying this expression, we get:
dθ/dx ≈ -0.00074675 radians per foot (rounded to five decimal places)
Therefore, the rate of change of the angle of elevation at x = 286 feet is approximately -0.00074675 radians per foot.