# Calculus

posted by .

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?

• Calculus -

A is angle in radians between perpendicular to shore at P (length 3 ) and light at shore (x which is 1 at start).
tan A = x/3
dA/dt = w = rad/sec = 4* 2 pi/60 = .419 rad/s
dx/dt = 3 d tan A /dt

dx/dt = (3/sec^2 A) dA/dt
dx/dt = 1.26 /sec^2 A = 1.26 cos^2 A

at x = 1, tan A = 1/3
so A = 18,43 deg
so cos A = .949
1.26 cos^2 A = 1.13 km/s

• Calculus -

Consider a right angled triangle,

tan(A)=x/3
cos(A)=3/sqrt(x^2+9)
sec^2(A)=(x^2+9)/3^2

Also,

x=3 tan(A)
dx/dt=3 sec^2(A) dA/dt
dx/dt=(x^2+9)/3 * 0.419

At x=1, dx/dt=1.396 km/s or 5027 km/hr.

The above attempt confused sec^2(A) with 1/sec^2(A), and 1/cos^2(A) with cos^2(A).

## Similar Questions

1. ### Calculus

A rotating beacon is located 1 kilometer off a straight shoreline. If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer …
2. ### Calc

A rotating beacon is located 1 kilometer off a straight shoreline. If the beacon rotates at a rate of 3 revolutions per minute, how fast (in kilometers per hour) does the beam of light appear to be moving to a viewer who is 1/2 kilometer …
3. ### Calc

A lighthouse is fixed 130 feet from a straight shoreline. A spotlight revolves at a rate of 11 revolutions per minute, (22 rad/min ), shining a spot along the shoreline as it spins. At what rate is the spot moving when it is along …
4. ### 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 …
5. ### 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 …
6. ### Calculus

A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes three revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?
7. ### calculus

a revolving light is located at 5 km from a straight shoreline with a constant angular velocity. what velocity does the light revolve if the light moves along the shoreline at the rate of 15km/min when the beam of light makes a angle …
8. ### Calculus 1

Related Rates: A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes two revolutions per minute. How fast is the beam of light moving along the shoreline when it is …
9. ### Calculus 1

A lighthouse is located on a small island 4 km away from the nearest point P on a straight shoreline and its light makes two revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?
10. ### Calculus

A light in a lighthouse 5 kilometers offshore from a straight shoreline is rotating at 4 revolutions per minute. How fast is the beam moving along the shoreline when it passes the point 5 kilometers from the point opposite the​ lighthouse?

More Similar Questions