calculus

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A searchlight rotates at a rate of 4 revolutions per minute.?
The beam hits a wall located 13 miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per hour) is this dot moving when the angle between the beam and the line through the searchlight perpendicular to the wall is pi/6? Note that dtheta/dt=4(2pi)=8pi.

  • calculus -

    velocity= dtheta/dt * distance

    in the units you have, with distance in miles, and dt in minutes, velocity will be in miles/minute

  • calculus -

    θπ

    cos(π/6)=sqrt(3)/2
    sec^2(π/6)=4/3

    dθ/dt=4 rev/min = 8π rad/min

    tan(θ)=x/13
    x=13 tan(θ)
    dx/dt=13 sec^2(θ) dθ/dt

    At θ=π/6

    dx/dt=13*(4/3)*8π=435.6 miles/min = 26138 miles/hr

  • calculus -

    Note to above attempt: velocity in x-direction, not radial velocity.

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