An object is located at the origin at time t=0. An elastic band connects the obj

Question

An object is located at the origin at time t=0. An elastic band connects the object to the point Q=(2,0). The object starts moving according to the parametic equations:

x(t)=t, y(t)= Square root of t

where t is time in seconds and the units on the axes are meters. The dotted line below shows the elastic band at a typical time t:

(a) The instantaneous rate of change of the length of the elastic band at time t=0 is ?

(b) The instantaneous rate of change of the length of the elastic band is 0 m/sec at time ?

Please show work, that'd be much appreciated. Thanks!

Explanation / Answer

s= sqrt((x-2)^2+y^2) ; x= t, y= sqrt(t)

ds/dt= ds/dx * dx/dt + ds/dy* dy/dt ==> ds/dt= (x-2)/sqrt((x-2)^2+y^2) * dx/dt+y/sqrt((x-2)^2+y^2)*dy/dt

since  x= t, y= sqrt(t)==> ds/dt= (t-2)/sqrt((t-2)^2+t)+ sqrt(t)/(t-2)/sqrt((t-2)^2+t)* 1/2sqrt(t)

==> ds/dt= (t-3/2)/(t-2)/sqrt((t-2)^2+t)

at t=0 ==> ds/dt= -3/4

if ds/dt= 0 ==> t= 3/2s

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