Answer :
Answer:
Step-by-step explanation:
How Fast is the Lighthouse Beam Moving?
Date: 10/13/2002 at 20:29:57
From: Leigh
Subject: Rates and derivatives
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 away from P?
Date: 10/15/2002 at 13:45:29
From: Doctor Roy
Subject: Re: Rates and derivatives
Hi,
Thanks for writing to Dr. Math.
Let's draw a quick picture to make the situation clearer:
3km
L ---------------P
\ |
\ |
\ |
\ | 1km
x \ |
\ |
\ |
A
Please note that the picture is not to scale. Point A is where
the beam of light hits the shore when it is 1 km away from P, and
x represents the length of the beam of light from the lighthouse.
We can find the value of x by using the Pythagorean theorem:
x^2 = 1^2 + 3^2
x^2 = 1 + 9
x^2 = 10
x = sqrt(10)
So, x is sqrt(10) km long, or roughly 3.16 km long.
Another fact we need to know is the rotational speed of the
lighthouse. The lighthouse beam makes 4 full revolutions in a minute.
So, we can express this as:
w = 4 rev/min
Do you remember the formula for the circumference of a circle? It is
given by:
C = 2*pi*r