   Chapter 2.8, Problem 24E

Chapter
Section
Textbook Problem

# A particle moves along the curve y = 2 sin ( π x / 2 ) . As the particle passes through the point ( 1 3 , 1 ) , its x-coordinate increases at a rate of 10  cm/s . How fast is the distance from the particle to the origin changing at this instant?

To determine

To find:

The rate of change of distance from particle to origin

Explanation

1. Concept:

Differentiate the given equation of curve with respect to time, and then use the distance of the particle from origin and differentiate to find the required rate.

2. Formula:

i. Derivative of sine :

ddx(sinx)=cosx

ii. Chain rule of differentiation:

ddtfx=ddxfx*dxdt

iii. Power rule of differentiation:

ddxxn=n*xn-1

3. Given:

i. Point ( 13,1 ),dxdt= 10

ii. Equation of curve,y=2 sin(πx2)

4. Calculations:

Equation is y=2 sin(πx2)

Differentiate it with respect to t using the sine and chain rule

dydt=2cosπx2*π2*dxdt

Substituting the values x=13 and dxdt=

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