Find an equation for θ in terms of x

Question

A boat is sailing towards a cliff that rises 272 meters above water level. Let θ be the angle of elevation from the boat to the top of the cliff, and let x be the distance from the boat to the cliff.

Find an equation for θ  in terms of x.

θ =

Find the rate of change of θ with respect to x when the boat is x=146 meters from the cliff. Round your answer to 3 decimal places.

Answer 1

tan (θ) = 272/x

So θ = tan^–1(272/x)

rate of change of θ

Let u = 272/x,

Using the chain rule: dθ/dx = d(tan^–1(u))/dx = [1/(1 + u²)](du/dx) =

Expressed in terms of x

1/(1 + u²) = 1/(1 + (272/x)²) = x²/(x² + 272²)

du/dx = –272/x²

dθ/dx = [x²/(x² +272²)](–272/x²) = (–272x²)/[x²(x² + 272²)] = –272/(x² + 272²) = –272/(x² + 73,984)

dθ/dx when the boat is 146 m from the cliff

–272/((146)² + 73,984) = –2.85 × 10^–3 rad/m = –0.164 °/m