Math 100 Assignment

Question

A searchlight rotates at a rate of 4 revolutions per minute. The beam hits a wall located 15 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 θ between the beam and the line through the searchlight perpendicular to the wall is π/6? Note that dθ/dt=4(2π)=8π.

Answer 1

in 15 seconds the light completes 1 revolution

4 revolutions per minute

1 revolution per 60/4 = 15 seconds

15 miles away

construct a right triangle with base = 15=b

altitude = a = 15tan30 = 15/sqr3 = 15sqr3/3 = 5sqr3= about 8.66

hypotenuse = 10sqr3 = about 17.32

the beam is 30 degrees = pi/6 radians from the perpendicular to the wall

dT/dt = change in Theta/time = 4(2pi)= 8pi, according to the given problem

or

dT/dt = 4(360)/60 = 4(6) = 24 degrees per second = 1440 degrees per minute

= 4(2pi) = 8pi radians per minute = 8pi/60 radians/second = 2pi/15 rad/sec

how fast is the beam moving along the wall?

about 17 miles per second

tanT = a/15 = opposite side over adacent

15tanT = a

a' = 15sec^2(T)T'

a' = 15sec^2(30)(dT/dt)

= 15(4/3)(24)

= 20(24)= 48 miles per second

= 48(60) = 2880 miles per minute

=2880(60) = 172,800 mph

since the speed does approach infinity as the angle approaches 90 degrees

a large rate of speed is possible, or likely

yet

there may be a large area in the above calculations

as a change from 29 to 31 degrees is about 17 m/s= 61,200 mph

if you find an error or two above, you get 2 or 3 points extra credit

3 months+ and no one else commented or posted a response?

that's tempting to try, even if the answers keep coming out

too different to be reliable

possible reasons for error(s) include mistaken conversions of units,

common arithmetic mistakes, trig mistakes,

or more

there have been some major, infamous, mistakes in rocket science

where the rocket gets astray in space due to the scientists getting

units mixed up. the metric system alone a major source of error for

american scientists. It happens. put sh in front of -it