[Guest]

Hi Forum,

I am to draw a graph representing the radius of a balloon as a function of time. Here is a given problem: "A man is inflating a spherical balloon by blowing air into the balloon at a constant rate. Draw a graph that best represents the radius of the balloon as a function of time."

I think I will let the x axis as the time and the y axis as the radius. Still don't seem to know how the line on the graph will look like though.

(y)

^

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----------------------------> x = time

[Aaryan]

Well, we can't say exactly how the line will look as we don't have any measurements, buy if I'm not mistiaken it is a direct variation function (in the form of y=kx

A general idea is that

a. It is a line

b. It crosses the origin

If we had numbers we could actually draw it, but that's a general idea.

[Guest]

Well you start with the function for the rate of change of the volume of the balloon.

V'=kt

Where V is the volume, k is the rate at which air is being blown into it, t is time.

This gives us V=(t^2)*k/2

But, when r is the radius, we also have this formula

V=3/4[pi]r^2 ([pi]=3.14159265...)

Therefore,

(t^2)*k/2=3/4[pi]r^2

and you can solve for either variable. Since the constants don't matter much we can say Ct=r.

I may have made an error in my calculations, so you might want to double-check them, but that's the process for solving the problem. Look at what formulas you have, and solve for what you need to solve for.

Edited May 15, 2011 by Chokfull

[Thalia]

Since the constant rate is not given in the problem, I don't think you can be exact. However, we know that the volume of a sphere (V)= 4pi(r³)/3. Let's say that the constant is 1 cubic inch per second (just for simplicity). At time (t)=0, the volume is going to be 0. Therefore, if we plug in 0 for V and solve for r, we get 0. If you do the same for t=1, 2, and 3, you get a graph that looks somewhat like a square root function.

Another way to look at it is this:

V= 4pi(r³)/3

V is also equal to t times the constant rate ©. V=tc

Therefore, 4pi(r³)/3=tc

Make up a number for the constant and replace c with that number.

Substitute different values for time and solve for r.

[Guest]

Since the constant rate is not given in the problem, I don't think you can be exact. However, we know that the volume of a sphere (V)= 4pi(r³)/3. Let's say that the constant is 1 cubic inch per second (just for simplicity). At time (t)=0, the volume is going to be 0. Therefore, if we plug in 0 for V and solve for r, we get 0. If you do the same for t=1, 2, and 3, you get a graph that looks somewhat like a square root function.

Another way to look at it is this:

V= 4pi(r³)/3

V is also equal to t times the constant rate ©. V=tc

Therefore, 4pi(r³)/3=tc

Make up a number for the constant and replace c with that number.

Substitute different values for time and solve for r.

Maybe the point of the problem is to find the generic function that relates the two variables. So instead of plotting points for arbitrary values, you could turn your equation into r = kt^a, where you don't know k, but you do know a, so you know the approximate shape of the graph.