Question

A searchlight rotates at a rate of 3 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 π4 ? note that dθ/dt=3(2π)=6π .

Answer 1

To find the speed of the dot of light on the wall, we can use the concept of angular velocity.

To find the speed of the dot of light on the wall, we can use the concept of angular velocity. The distance covered by the dot of light on the wall is equal to the arc length on the wall. We can find the arc length using the formula:

Arc Length = radius × angle

Given that the radius of the wall is 15 miles and the angle θ is π/4 radians, we can calculate the arc length. Since the searchlight rotates at a rate of 3 revolutions per minute, the angular velocity (dθ/dt) is 6π radians per minute. Converting this angular velocity to miles per hour, we get:

Velocity (in miles per hour) = Arc Length (in miles) × (angular velocity (in radians per minute) × 60) / 2π

By substituting the values, we can calculate the velocity of the dot of light on the wall.