《AP Calculus》A plane flies horizontally at an altitude....Find the rate of the plan travels at some time

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Question:

A plane flies horizontally at an altitude of 5 km and passes directly over a tracking telescope on the ground. When the angle of elevation is π/3, this angle is decreasing at a rate of π/6 rad/min. How fast is the plane traveling at that time?

Solutions:

這是典型的 Related rate problem,答題方式很簡單,基本上有一套SOP

  1. 首先先把題目的資訊整理清楚,畫出圖形來標記
  2. 列出相關的算式,比如面積、體積、長度等
  3. 把算式對時間作微分(Differentiate the equation on both side, usually by t)
  4. 帶入已知的資訊就可以得到答案

我們來看看這題怎麼做,首先把資訊整理好,並且畫出圖形,我們有

A typical related rate question. Ap Calculus AB BC

 

 

 

 

 

 

 

 

 

題目告訴我們 an altitude of 5 km,意思是 h=5,我們假設水平移動的距離為y,this angle is decreasing at a rate of \frac{\mathrm\pi}6 rad/min,這句話又告訴我們

\frac{d\theta}{dt}=-\frac{\mathrm\pi}6(rad/min) 因為是decreasing所以要記得負號

首先先從圖形上的關係得到

\frac y5=cot\theta

把算式對時間作微分得到

\begin{array}{l}\frac{dy}{dt}=5\left(-csc^2\theta\right)\times\frac{d\theta}{dt}\\\\\end{array}

\frac{dy}{dt}=5\left(-csc^2\frac{\mathrm\pi}3\right)\left(-\frac{\mathrm\pi}6\right)=\frac59\mathrm\pi

所以我們就得到飛機在這個角度時的速度是 {{\frac{dy}{dt}}=}{\frac{5\mathrm\pi}9}

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