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Answer 1 · 2018-02-04T16:00:01

$sf(("d"theta)/dt=- 0.02color(white)(x)" rad/s")$

In degrees:

$sf(("d"theta)/dt=-1.15^@"/s")$

MFDocs

We are told:

$sf((dX)/dt=8color(white)(x)"ft/s")$

We need to find $sf(("d"theta)/dt$

From the geometry of the situation, Pythagoras tells us:

$sf(X^2+100^2=200^2)$

$sf(X^2+10,000=40,000)$

$sf(X=sqrt(30,000)color(white)(x)ft)$

and

$sf(sintheta=100/200=0.5)$

$sf(theta=30^@)$

At this instant:

$sf(tantheta=100/X)$

Differentiating implicitly with respect to time t :

$sf(d/dt[tantheta]=d/dt[100/X])$

$sf(sec^2theta.("d"theta)/dt=-100/(X^2).(dX)/dt)$

$sf(("d"theta)/dt=-(100cos^2theta)/(X^2).(dX)/dt)$

Putting in the numbers:

$sf(("d"theta)/dt=-100xxcos^2(30)/(30,000)xx8)$

$sf(("d"theta)/dt=-100xx0.75/(30,000)xx8=-0.02color(white)(x)"rad/s")$

$sf(2picolor(white)(x)"rad"=360^@)$

$:.$ $sf(1color(white)(x)"rad"=360/(2pi)=57.3^@)$

$:.$ $sf(0.02color(white)(x)"rad"=57.2xx0.02=1.15^@)$

In degrees:

$sf(("d"theta)/dt=-1.15^@"/s")$