Can anyone help me with Related Rates? PLEASE?

Question

A weather balloon rises through the air at a rate of 500m/min. Every 1000m, the decrease of air pressure outside the balloon causes its radius to increase by 8cm. How rapidly is the volume increasing at the instant the radius is 90cm?
i know the answer, but i just dont know how to get it..please show me HOW to do it! THANKS!

Answer 1

Assume spherical approximation for balloon:

Volume of weather balloon at time, t is:

V(t) = 4/3 pi r(t)^3

(1).....dV(t)/dt = 4 pi r(t)^2 x dr(t)/dt (chain rule)

let t be in minutes

r(t)= .04t m, that is (500/1000)*0.08*t m

d r(t)/dt = .04m/min

So when r(t) = 90cm = .9m, we have from (1)

d V(t)/dt = 4 pi (0.9)^2 x .04 m^3/min

=0.4072 m^3/min

Answer 2

A fun problem. Try to break this down into smaller parts. And ignore numbers.

A weather balloons radius INCREASES as it RISES, thereby INCREASING the volume.

O.K. What are we trying to find? The rate of change of volume of the balloon (dV/dt). Well then, we will of course need an equation for the volume of the balloon.

Assuming the balloon is a sphere,

V=4/3*pi*r^3

O.K., like I said put blinders on and just look at the equation above. We can find dV/dt simply by differentiating with respect to time. DO NOT forget the CHAIN RULE!!! Also, keep in mind that we know the radius r is function of time.

dV/dt = 4*pi*r^2*dr/dt

O.K. we have one unkown which is dr/dt, the rate of change of the radius.

Now we look back up to the problem and see dr/dt is a function of the height of the balloon above the earth. You can find dr/dt by one of two ways (whichever way is easier to visualize).

One way is to just find dr/dt by units....

For instance, we know the units of dr/dt are length per time. We also know that the change in radius (dr) for what we are looking at is 8 cm. The only number with time (dt) is the 500 m/min. We want time of bottom so multiply 8 cm * 500 m/min, which is,
4000 cm*m/min

There is a problem because the top is length squared, we just want length. But we know that the above number occurs every 1000 m.... so divide by 1000m and.... dr/dt = 4cm/min.

Therefore dV/dt = 4*pi*(90 cm)^2*4 cm / min

Multiply out and verify that you have (AND WRITE DOWN as part of the final answer) correct units.

Another way to look at dr/dt is to simply say, ok, our radius increases 8 cm every 1000 m, we can go 500 m/min which means that we can go 1000m every TWO minutes. This means in two minutes our radius increases 8 cm. Therefore dr/dt = 8/2 = 4 cm/min.

Enough said.
Calculus, until you get used to it, is a very piecemeal process. The biggest calculus part is actually the easiest part. V= whatever, so dV/dt = the derivative of whatever with respect to time.

Try to go as far into the problem as you can WITHOUT sticking in numbers... usually this is much clearer and you will even be able to see some relationships between variable better.

And keeping track of units is crucial.

Answer 3

Let's see:

We're trying to find dV/dt - the time rate of change of volume.

What do we know?

Assuming that the balloon is spherical, the volume of a balloon having radius r is 4/3*pi*r^3.

If the balloon is rising at a rate of 500 meters per minute, and every 1000 meters the balloon's radius increases by 8 cm, then:

500 meters/minute * 8 cm/1000 meters = 4 cm/minute

This is the time rate of change of the radius.

We can find dV/dt as being equal to dV/dr * dr/dt.

Find dV/dr by taking the derivative of the volume formula with respect to radius:

V(r) = 4/3*pi*r^3
dV/dr = 4*pi*r^2
dr/dt = 4 cm/minute

dV/dt = dV/dr * dr/dt
dV/dt = 4*pi*(90cm)^2 * 4 cm / minute
dV/dt = 4*pi*(8100 cm^2) * 4 cm / minute
dV/dt = 129600*pi cm^3 / minute, or 0.1296*pi m^3 / minute


It's been a long time since I've done one of these, but I think that's right.

Answer 4

Lets call V = volume (m^3), r = radius (m), h = height (m), t = time (min).
Now, write down what we know:
dh/dt = 500. dr/dh = 0.08/1000 = 0.00008. V = 4/3 πr^3. We want to know dV/dt.

Differentiate the volume equation to get dV/dr = 4πr^2.
Now, dV/dt = dV/dr * dr/dh * dh/dt, since everything cancels out except what we want. So dV/dt = 4πr^2 * 0.00008 * 500. Substitute in r = 0.9, to get 0.40715 m^3 / min.