Question

RELATED RATES QUESTION!!

A rotating beacon is located 4 miles out in the water. Let A be the point on the shore that is closest to the beacon. As the beacon rotates at 8 rev/min, the beam of light sweeps down the shore once each time it revolves. Assume that the shore is straight. How fast is the point where the beam hits the shore moving along the shore at an instant when the beam is lighting up a point 4 miles along the shore from the point A?

Answer 1

Hint: Because this is rotational motion the velocity will be the (8)circumference. Multiply this by 60 if you want this in miles per hour.

Now for the related rate. The distance from the beacon to the shore changes as it rotates therefore the velocity changes. How is the radius going to change as this beam rotates. You can find this by using the Pythagorean formula. I hope this helps.

Answer 2

actually... there's an easier way to do this:

Answer 3

differentiate with respect to time, find theta when x=4, plug that and d(theta)/dt (in rad.s/sec) into the expression for dx/dt

Answer 4

any questions?

Answer 5

i'm not getting it

Answer 6

which part?

Answer 7

the sketch? the differentiation? finding and plugging in theta and d(theta)/dt?

Answer 8

making the equation and differentiating it
i'm not sure where theta comes in, is it the rev/ min?

Answer 9

but they don't give you a theta so you can't derive it with theta

Answer 10

x= 4 tan(theta) is the equation...
it's just trig.

Answer 11

theta we can find ...
(but don't use it until *after* you do the differentiation)

...because they ask for dx/dt when x=4

Answer 12

so i get dx/dt = (dtheta,dt)4tan(theta)

Answer 13

*dtheta/dt

Answer 14

use the chain rule theta is a function of time:
(x= 4tan( f(t) ) )'

Answer 15

what's the derivative of tan(x)

Answer 16

sec^2x

Answer 17

yes. so what's the derivative of tan( f(x) ) ?

Answer 18

sec^2 ( f(x) ) * f ' (x)

Answer 19

so sec^ 2 (theta) * d(theta)/dt (whole thing times 4)

Answer 20

dx/dt = 4*sec^2(theta) * d(theta)/dt

Answer 21

now we need to find theta when x=4 and we need to find d(theta)/dt in appropriate units.

Answer 22

what's theta going to be (from the sketch) if x=4 ?

Answer 23

sqrt. 2/ 2

Answer 24

45 degrees

Answer 25

:)

Answer 26

now for d(theta)/dt

Answer 27

you want your answer in miles per hour?

Answer 28

if you want your answer in miles per hour, convert the RPM to rad.s/hour...

Answer 29

hello?

Answer 30

480rev/hr

Answer 31

so 480= 4 (pi/4)^2 d(theta)/dt

Answer 32

480 rev/hour * 2pi rad / rev = 960 pi rad.s / hour = d(theta) /dt

Answer 33

so dx/dt =4*sec^2(45) *960pi

Answer 34

the answer is supposed in rads/ min

Answer 35

I just asked that...

Answer 36

so why look for dx/dt