Chapter 4, Problem 58RE

To determine

To find : How fast the particle approaching the origin as it passes through the point $(5,12)$ .

Answer to Problem 58RE

The answer:

Approaching at $5 \text{m}/\text{sec}$ .

Explanation of Solution

Given information:

The differential functions are:

  $\begin{array}{c}dx/dt=-1 \text{m}/\text{sec }\\ dy/dt=-5 \text{m}/\text{sec}\end{array}$

Calculation:

The particle is now travelling in the arrow's direction and is located at $(5,12)$ . A line connecting the particle and the origin is denoted by $h$ .

To find $\frac{dh}{dt}$:

  

At the moment,

  $\begin{array}{c}x=5,\\ y=12\\ h=\sqrt{5^2+{12}^2}\\ =13\end{array}$

Implicitly differentiate with respect to $t$ , plug in all the known values, then , solve for $\frac{dh}{dt}$ :

  $\begin{array}{c}{h}^2=x^2+y^2\\ \Rightarrow 2h\frac{dh}{dt}=2x\frac{dx}{dt}+2y\frac{dy}{dt}\\ \Rightarrow 2(13)\frac{dh}{dt}=2(5)(-1)+2(12)(-5)\\ \Rightarrow \frac{dh}{dt}=-130/26=-5\end{array}$

Therefore, the required approaching at $5 \text{m}/\text{sec}$ of the origin as it passes through the point $(5,12)$ .