A searchlight rotates at a rate of 2 revolutions per minute. The beam hits a wall located 12 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/5]?. Note that dθ/dt=2(2π)=4π.

I understand that I need to use tangent here but I can't get farther than that. Thanks in advance! @Mathematics

Question

A searchlight rotates at a rate of 2 revolutions per minute. The beam hits a wall located 12 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/5]?. Note that dθ/dt=2(2π)=4π.

I understand that I need to use tangent here but I can't get farther than that. Thanks in advance!   @Mathematics

amistre64 · Answer

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amistre64 · Answer

youre right about the tangent function to relate the motion and the angle speed

amistre64 · Answer

tan(a) = w/12

take the derivative of this with respect to t, for time

sec^2(a) a' = w'/12  and solve for w', for wall speed

amistre64 · Answer

i got a' = d/dt theta in this

amistre64 · Answer

so, 12 * sec^2(pi/5) * 4pi = w'