# calculus

posted by .

A searchlight rotates at a rate of 4 revolutions per minute. The beam hits a wall located 9 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 5? Note that ddt=4(2)=8

• calculus -

let x be the angle
let d be the distance of the dot from the point on the wall closest to the light

tan(x) = d/9
sec^2(x) dx/dt = 1/9 dd/dt
dd/dt = 1/9 sec^2(x) dx/dt

dx/dt = 4*2pi/min
I assume x = 5°, since 5 radians would point away from the wall.

dd/dt = 9 * 1.00765 * 25.13274 = 227.843 mi/min * 60min/hr = 13670.6 mi/hr

Check for sanity. If the wall were a circle 9 miles in radius, the circumference of the wall would be 9*2pi = 56.5 miles. That distance would be covered 4 times per minute, making the dot travel at a constant speed of 13572 mi/hr.

That would be the slowest speed observed when traveling along a straight wall, at the instant when the dot is closes to the light. Since 5° is a small angle, we'd expect the speed to be close to that figure, and it is, but slightly faster.

## Similar Questions

1. ### calculus

Consider the illustration, which shows a rotating beam of light located 0.5 mile from a shoreline. The beam rotates at a rate of 4 revolutions per minute. How fast (in miles per minute) is the distance between the beam and the point …
2. ### Calculus

A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline. Its light makes 3 revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1.7 kilometers from …
3. ### calculus

A searchlight rotates at a rate of 4 revolutions per minute.?
4. ### calculus

A searchlight rotates at a rate of 4 revolutions per minute.?
5. ### Calculus

A searchlight rotates at a rate of 2 revolutions per minute. The beam hits a wall located 8 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 …
6. ### Calculus

Consider the illustration, which shows a rotating beam of light located 0.5 mile from a shoreline. The beam rotates at a rate of 4 revolutions per minute. How fast (in miles per minute) is the distance between the beam and the point …
7. ### Calculus

A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?
8. ### calculus

A searchlight rotates at a rate of 3 revolutions per minute. The beam hits a wall located 7 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 …
9. ### calculus

a laser printer is placed on a platform that rotates at a rate of 20rev/min. the beam hits a wall 8m away producing a dot of light that moves horizontally along the wall. let tetabe the angle between the bean and producing the line …
10. ### calculus

A patrol car is parked 50 feet from a long warehouse The light on the car turns at a rate of 30 revolutions per minute. How fast is the light beam moving along the wall when the beam makes the fol­ lowing angles with the line perpendicular …

More Similar Questions