Question

Can anyone help me with Calculus?

A rescue boat uses a long towline and a winch located twelve feet off the face of
the water to receive a small yacht in distress. on the rescue boat the towline is wrapped
once around a 3 - foot radius cylinder that retrieves the line by turning at a rate of 2
revolutions per minute. Determine the rate at which the yacht is approaching the tugboat
when it is 50 ft away .

Answer 1

Would I use the formula for volume of a cylinder or use Pythagorean theorem?

Answer 2

pythagorean theorem

Answer 3

the cylindrical shape is irrelevant

Answer 4

b = 12

Answer 5

dy/dt is 3-foot?

Answer 6

3 is not important

Answer 7

a = 50

Answer 8

c = sqrt(a^2+b^2)

Answer 9

you want to find da/dt

Answer 10

is there a constant?

Answer 11

c = sqrt(a^2+b^2) -> c^2 = a^2 + b^2 -> c^2 - b^2 = a^2

Answer 12

*derivative*
2c(dc/dt) - 2b(db/dt) = 2a(da/dt)

Answer 13

db/dt = 0, so

2c(dc/dt) - 2b(0) = 2a(da/dt) -->2c(dc/dt) = 2a(da/dt) -> c(dc/dt) = a(da/dt)

Answer 14

is it sqrt 2644?

Answer 15

2c(dc/dt) = 2a(da/dt) -> (da/dt) = c/a *(dc/dt)

Answer 16

hypotenuse is sqrt2644

Answer 17

just trig

Answer 18

The ? determines the length of the tow line to the yacht. You will need to determine the circumference of the cylinder as that will give yu the number of feet of tow line that is being pulled in each minute.

Answer 19

how do i know which one is dx/dt...

Answer 20

(da/dt) = c/a *(dc/dt)
This is where the cylinder comes in; for dc/dt we know 2 revolutions per min, which means total length of rope pulled is 2*[2pi(3)] = dc/dt

Answer 21

x is a

Answer 22

dc/dt = 12pi

Answer 23

da/dt = c/a *(dc/dt) = sqrt2644 / 50 * 12pi

Answer 24

da/dt = dx/dt

Answer 25

x=a
y=b

Answer 26

TLDR; dx/dt = sqrt2644 / 50 * 12pi

Answer 27

how would I know da/dt=c/a*dc/dt

Answer 28

from differentiating the pythagorean theorem with respect to t

Answer 29

(da/dt) = c/a *(dc/dt)
This is where the cylinder comes in; for dc/dt we know 2 revolutions per min, which means total length of rope pulled is 2*[2pi(3)] = dc/dt

how did u get that from 2revolutiongs per min?