Find the rate of change of the angle of elevation dθ/dxx when x = 275.

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.

dθ⁄dx when 275.

275

Answer 1

Find the rate of change of the angle of elevation dθ/dxx when x = 275.

A building that is 225 feet tall casts a shadow of various lengths 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, as seen in the figure above.

dθ⁄dx when x=275.

θ'(275)=_________ 

Hello Matt,

( ⊗ ) <--- Sun

..... Building = 225 ft. tall

...... ⇓

.....⌈—⌉...\

.....⌈—⌉......\

.....⌈—⌉.........\

.....⌈—⌉............\

.....⌈—⌉...............\

.....⌈—⌉.............. θ \ <--- Angle of elevation (θ)

__________________

⇑ Shadow cast is 275 ft. along the ground, represented by the grey line shown above.

Note: As the sun rises higher in the sky, from behind the building, the shadow made will retreat towards the building and become shorter and shorter until about noon, when the sun is directly overhead and there is no shadow.

Use a tangent function to set up the problem, since the side opposite to the angle θ, is the height of the building, which is a constant 225 ft. And, the side adjacent to the angle θ, is the x-axis or the shadow length, which is not a constant and will be changing as a function of the angle θ:

Using:

tanθ = opp / adj

tanθ = 225 / x

Take the derivative of the function with respect to x as follows:

d/dx [ tanθ = 225 / x ]

d/dx [ tanθ = 225x^-1 ] ......* Important to put dθ/dx after the derivative of tanθ, because it is not a function of x.

sec²θ * dθ/dx = - 225 / x² ..........* Solve for dθ/dx in order to get the rate of change of the angle θ, shadow length x, with respect to the change in the.length x,

dθ/dx = - 225 / (x² * sec²θ) .........* Substitute x = 275 ft and also solve for θ using:

tanθ = 225 / 275

θ = tan^-1 (225 / 275)

θ = 0.68573 radians, or in degrees θ = 39.28941º

Finally, substitute the x and θ back into the dθ/dx expression, and solve:

dθ/dx = - 225ft / (x² * sec²θ)

dθ/dx = - 225ft / ((275ft)² * sec²(0.68573))

dθ/dx = - 225ft / ((75625ft² * 1.66942)

dθ/dx = - 225ft / (75625ft² * 1.66942)

dθ/dx = - 0.0017822 rad/ft

Answer: The angle of elevation is decreasing at a rate of 0.001782 radians per foot of added shadow length, when the building casts a 275 ft shadow.