How do you find the rate that the angle from the car's starting position to its current position as it's increasing?

Question

A car starts driving from a point 18 miles north of a city center, driving due east. After a while, the car is 80 miles east of where it started, and is driving at a rate of 60 miles per hour. Find the rate that the angle from the car's starting position to its current position, measured from the city center, is increasing.

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Answer 1 · 2018-03-19T06:54:57

#(d theta)/dt=0.1606# radians per hour of increase

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At the moment specified in the problem,

#x=80# miles

#dx/dt=60# miles/hour

#tantheta=x/18=80/18=40/9#

Let's take the derivative of the above function:

#sec^2theta d theta=1/18dx#

Let's divide both sides by #dt#:

#sec^2theta (d theta)/dt=1/18*dx/dt#, #color(red)(Equation - 1)#

At the specified moment.

#sec^2theta=1+tan^2theta=1+1600/81=1681/81#

Let's substitute known values in #color(red)(Equation - 1)#:

#1681/81((d theta)/dt)=1/18*60=10/3#

#(d theta)/dt=(10/3)/(1681/81)=10/3*81/1681=270/1681#

#(d theta)/dt=0.1606#

The angle is increasing at approximately #0.1606# radians per hour at the specified moment.