Writing equations for rate of change problems

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SUMMARY

The discussion focuses on deriving an equation to calculate the rate of change of the angle of a telescope positioned 75 meters above water level as a boat approaches at 6 m/s. The correct approach involves using the equation tan(θ) = 75/x, where θ is the angle of elevation. Differentiating this equation with respect to x using the chain rule leads to the necessary calculations for determining the rate of change of the angle.

PREREQUISITES
  • Understanding of trigonometric functions, specifically tangent and arctangent.
  • Knowledge of differentiation techniques, including the chain rule.
  • Familiarity with related rates in calculus.
  • Basic concepts of angle measurement in right triangles.
NEXT STEPS
  • Study differentiation techniques, focusing on the chain rule in calculus.
  • Learn about related rates problems in calculus to apply similar concepts.
  • Explore trigonometric identities and their applications in calculus.
  • Practice solving problems involving angles of elevation and depression in real-world scenarios.
USEFUL FOR

Students studying calculus, particularly those focusing on related rates and trigonometric applications, as well as educators looking for examples to illustrate these concepts.

Poppynz
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Hi
Im trying to write an equation for the question below. i did write one but am pretty sure it is wrong because I need to differentiate arctan in it and we have not been taught that yet. Could someone please point me in the right direction with writing it?

Homework Statement




A telescope is 75m above water level on a cliff and a boat is approaching at 6m/s. what is the rate of change of angle of the telescope when the boat is 75m from shore.



Homework Equations





The Attempt at a Solution



I thought it might be y=arctan(75/x) because the angle specified is found using arctan(opposite/addjacent).

Any suggestions appreciated :)
 
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Poppynz said:
… I thought it might be y=arctan(75/x) because the angle specified is found using arctan(opposite/addjacent).

Any suggestions appreciated :)

Hi Poppynz! :smile:

Just write it tanθ = 75/x, and differentiate both sides wrt x …

from the chain rule, you'll get a sec2θ, which you can rewrite in terms of the original 75/x :wink:
 
Thanks that seems much easier :)
 

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