Question

Related Rates: I know I have to do derivatives but I'm stuck on how to get from being given dv/dt to finding area. Also confused about finding radius when not given dr/dt only the height
A chemical tank is leaking a toxic substance at a rate of 100cm^3/minute. Slick is semi-circ. in shape and has a thicknesss of .5cm
a) After 1 hour how big an area does the slick cover?
b) At 1 hour, at what rate is the radius of the slick changing?

Answer 1

i think if we just half the height, we can create a full circle model

Answer 2

V = pir^2 h/2

V' = 2pi r r' h/2 + pi r h'/2 ; but h' = 0 since the height is not changing

V' = pi r r' h

100 = .25 pi r r'

400/pi = r r'
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area covered is just the base of the circle regardless of height

A = pi r^2

A' = 2pi r r' ... per minute, we already have determined the r r' = 400/pi

A' = 2pi(400/pi) = 800 cm^2 per minute

Answer 3

use 60 minutes to determine the area after an hour

use the Area at an hour to determine the radius, and solve for r' in the V' setup

Answer 4

might have some errors in calculations ... but the concept should be sound

Answer 5

im thinking i mentally halved the height to begin with and it should be:

100 = .5 pi r r'

200/pi = r r'