A light house is 1 km off shore. A marathoner is running along the shore. The search light on the light house is rotating at a rate of 1/6 revolution per minute when the runner is 3 km from the point on the shore that is the closest to the light house.The runner is illuminated by the beam. How fast is the beam of the light moving along the shore at this moment?
velcity=1/6 rev/min * 3km*2PIrad/rev
thank you very much 4 ur reply.But can u tell me how did u get this answer.
I have problems converting fractions to decimals. Can you help me.
To solve this problem, we need to use related rates, where we find the rate of change of one quantity with respect to another. In this case, we want to find how fast the beam of light is moving along the shore.
Let's set up a right triangle to represent the situation. The runner's position on the shore is one of the legs, the position on the shore closest to the lighthouse is the other leg, and the line connecting the runner to the lighthouse is the hypotenuse.
Let's call the distance between the runner and the closest point on the shore "x". Since the runner is 3 km from the closest point on the shore, we have x = 3 km.
The hypotenuse of the triangle represents the light beam, and it is always one kilometer long, as stated in the problem.
Now, we can express the relationship between the quantities involved. By the Pythagorean theorem, we have:
x^2 + 1^2 = (3 + 1)^2
x^2 + 1 = 16
x^2 = 15
Differentiating implicitly with respect to time, we have:
2x(dx/dt) = 0
Since x = 3 km, we have:
2(3)(dx/dt) = 0
6(dx/dt) = 0
dx/dt = 0
Therefore, the beam of light is not moving along the shore at this moment.