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

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:

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

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

$\frac y5=cot\theta$

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

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

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