Question

The radius of a sphere is increasing at a rate of 4mm/s. How fast is volume increasing when the diameter is 80mm.

Answer 1

You start off knowing dr/dt and you want to find out dv/dt

Answer 2

To get that you can just do \(\Large \frac{dr}{dt}*\frac{dV}{dr}\)
Do you know how to get dV/dr?

Answer 3

dV would just be the derivative of the volume correct?

Answer 4

Well, tempest, we're dealing with related rates, so it's not enough anymore to say just "the derivative." We have to say "derivative with respect to..."
dV/dr means derivative of volume with respect to the radius.
dV/dt means derivative of volume with respect to time.

Answer 5

Ok. I have the solution to the problem and I am confused on how they simplified dV/dt
It is number 6

Answer 6

What they have there is
dV/dt = dV/dr * dr/dt

Answer 7

The diameter is 80, so for the radius, they plugged in r = \(\frac{1}{2}*80\)

Answer 8

Because the radius is half. Ok I get that

Answer 9

and \(\Large \frac{4}{3}\pi*3\) became \(\Large 4\pi\)
And of course, the problem told you that dr/dt was 4, so they plugged 4 in for dr/dt.

Answer 10

So basically, I just needed to find the derivitive of the volume with respect to time, and multiply that with the deriv of the radius with respect to time?

Answer 11

Because the equation of the volume has radius in ti so that is how they are related?

Answer 12

What you said just now is... not quite right. The problem asks you for the derivative of the volume with respect to time, but it's not so simple to just find that.
However, we know that V = (4/3)pi*r^3
so, from that equation, it's easy to find derivative with the volume with respect to RADIUS.

Answer 13

Once we know derivative of the volume with respect to radius, we can multiply that by dr/dt to get dV/dt like we wanted.

Answer 14

How do we get the deriviative of dV/dr

Answer 15

You have the equation
V = (4/3) pi*r^3
Take the derivative with respect to r.

Answer 16

Use exponent rule.

Answer 17

v' = 3r^2? I am unsure about the (4/3)pi

Answer 18

That just gets multiplied by the 3.

Answer 19

derivative of ax^n
is
an*x^(n-1)

Answer 20

So no need to take the deriviative of 4/3 pi because it will get multiplied by 3

Answer 21

That part of the equation is just a constant.

Answer 22

ok

Answer 23

So dV/dr = (4/3)pi*3r^2 = 4pi*r^2

Answer 24

That is what we are solving for right? I think i am making this harder then it needs to be. I have been stumped on this problem for a good 20 minutes...

Answer 25

lol well perhaps I'm not explaining it super simply.

Answer 26

The radius of a sphere is increasing at a rate of 4mm/s. How fast is volume increasing when the diameter is 80mm.

"Radius of a sphere is increasing at a rate..."
dr/dt = 4
"How fast is volume increasing?"
We need to find dV/dt

\(\Large \frac{dr}{dt}*\frac{dV}{dr} = \frac{dV}{dt}\)
So we have a way to find it, but we must first find dV/dr. How do we find dV/dr?
V=(4/3)pi*r^3
Derive with respect to r.

Answer 27

So since dr/dt = 4 and dV/dt = 4pi*r^2, I just plug in 4 into r?

Answer 28

Nonono. dV/dr = 4pi*r^2
You haven't found dV/dt yet.

Answer 29

Ok so how do we find it?

Answer 30

\(\Large \frac{dr}{dt}*\frac{dV}{dr} = \frac{dV}{dt}\)

Answer 31

oh... dr/dt is 4 and dV/dr is 4pi r^2. We are solving for when the dimaeter is 80. So r is = 40 and then * dV/dr by dr/dt

Answer 32

when dr/dt is 4

Answer 33

Right, I believe.

Answer 34

Ok i got it correct.

Answer 35

So basiclly, find a formula that related both derivitves, and derive that formula respect to one of the varibles. ex: dx/dt and dy/dt. You need to relate using a formula derivitive dy/dx

Answer 36

Can someone confirm?