Question

Suppose oil spills from a ruptured tanker and spreads in a circular pattern. If the radius of the oil spill increases at a constant rate of 3m/s, how fast is the area of the spill increasing when the radius is 40m?
Answer in m^2/s

Answer 1

@thomaster @robtobey @phi

Answer 2

\[ \frac{d}{dt}\left(A= \pi r^2 \right)\\
\frac{d}{dt}A= \pi \frac{d}{dt}r^2\\
\frac{dA}{dt}= 2 \pi r \frac{dr}{dt}
\]
radius of the oil spill increases at a constant rate of 3m/s means dr/dt = 3 m/s
when the radius is 40m? i.e. r= 40 m
find dA/dt

Answer 3

i dont get how to find it.

Answer 4

dA/dt is the change in area with respect to time.
dA/dt is how fast is the area of the spill increasing

on the right side of the equation, you know all of the values.

Answer 5

the right side would be \[6\pi r\]?

Answer 6

how fast is the area of the spill increasing when the radius is 40m?

Answer 7

i dont know. I dont get this

Answer 8

the idea is not too tricky, but sometimes things don't "click"

but in this case, r is the radius. they want dA/dt when r= 40 m, and dr/dt= 3 m/s

Answer 9

so when i plug in dA/dt and dr/dt into the equation am i supposed to put it as 40 and 3 or as something else

Answer 10

You have an equation
\[ \frac{dA}{dt}= 2 \pi r \frac{dr}{dt} \]
you know r=40 and dr/dt =3
replace r with 40 and dr/dt with 3
simplify to find dA/dt

Answer 11

so its 240pi

Answer 12

or about 754 m^2/sec

Answer 13

ok

Answer 14

how would you do this for any other situation

Answer 15

the idea is if you know (in a very simple example)

y = x

and x changes, then y changes.
we can take the derivative with respect to time:
dy/dt = dx/dt
that says that if the change in x is dx/dt (think of this as a rate)
then the change in y will be the same rate. This should make sense, because y=x
and if x is changing at 1 step per second, y will be changing at 1 step per second.

Answer 16

if
y= 2x
if x changes at 1 step per second, y will go up by 2 steps per second

take the derivative with respect to time
dy/dt = 2 dx/dt

that says the change in y is twice as big as the change in x. in other words, if x goes up by 1 step every second, y will be going up at 2 steps every second.

Answer 17

So with a problem like this: The length of a rectangle is increasing at a rate of 8cm/s and its width is increasing at a rate of 5cm/s. When the length is 30cm and the width is 25cm, how fast is the area of the rectangle increasing? how would you do it

Answer 18

which derivatives do you have to take. the length and width?

Answer 19

you start by writing down an equation that has area , length and width in it.

Answer 20

I assume you know
A= L*W

Answer 21

yup

Answer 22

in your problem, all three A, L and W are changing, so we treat all three as a variable

step 1: write down the the equation
A = L W
step 2: take the derivative with respect to time
dA/dt = d/dt(L W)
on the right side, that is the derivative of two variables... we use the product rule
d(u v) = u dv + v du

Answer 23

can you take a stab at it?
dA/dt = d/dt(L W)

Answer 24

what would d/dt be?

Answer 25

d/dt means take the derivative with respect to t

it is like d/dx except we have t instead of x

we use the normal rules: d/dt of t = dt/dt = 1
d/dt (y) = dy/dt
d/dt (x) = dx/dt

d/dt t^2 = 2t dt/dt = 2t

follow ?

Answer 26

ok so what are we taking the derivative of?

Answer 27

dA/dt = d/dt(L W)

do the right-hand side. L and W are variables (just like x and y)

Answer 28

so it would be L(dW/dt)+(dL/dt)W?

Answer 29

yes. looks good.

As an aside, In physics people would write \( \dot{W} \) rather than dW/dt (Newton used this version) because it is easier. But it isn't easier to type. sometimes people use W' for dW/dt but it is never obvious that it is dW/dt (as opposed to dW/dx for example)

Answer 30

i havent done physics yet. only in ninth grade

Answer 31

you started with
A = L W
took the derivative with respect to time and found
dA/dt = L dW/dt + W dL/dt

now if we know the stuff on the right hand side, we can compute dA/dt

Answer 32

The problem gives all the info we need

Answer 33

L=30 W=25 dW/dt=5 dL/dt=8

Answer 34

150+200=350?

Answer 35

yup i got it. thanks so much. you are an amazing helper

Answer 36

yes. But it is a good habit to keep track of the units. dA/dt = 350 cm^2/sec

Answer 37

ok thanks so much!!!!!!