Expert: Scotto - 1/23/2011

Question
QUESTION: How fast is the surface area of a spherical balloon increasing when the radius is 10cm and the volume is increasing at 15 cm^3/sec?

Thank you so much. I do not even know how to begin this question

ANSWER: It is known that the surface area is A = 4*pi*r^2.
Since we want the change with respect to time of area.
That is, find dA/dt if A(t) = 4*pi*r^2(t) given dV/dt = 15cm^3/sec.

Now V(t) = 4*pi*r^3(t)/3, and we know dV/dt = 15cm^3/sec.
It can be seen that dV/dt = 4*pi*r^2(t)(dr/dt).

Now dV/dt = 15 and r(t) = 10 cm, so we have 15 = 4*10^2(dr/dt).
That is the same as 15 = 400(dr/dt), so dr/dt = 15/400 = 3/80.

The rate of change of the surface area is dA/dt = 8*pi*r(t)*(dr/dt).
Since r(t) is given as 10 and dr/dt was just found as 3/80, we can find dA/dt,
and that's what the problem is looking for.


---------- FOLLOW-UP ----------

QUESTION: How do you produce a formula for dV/dt = something x dr/dt. What is that something ratio. thats what i cannot get

Answer
Since V and r are both functions of t, both sides are differentiated with respect to t.

Definition of derivatives
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Derivatives, when expressed as dF/dx, means the derivative of the function F in terms of x.
The derivative of x(t) is dx/dt, or the derivative of x in terms of t.
In this way, dF/dt = (dF/dx)(dx/dt), since the dx's cancel in fractional arithmetic.

Example: y = x^3
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This is done in calculs when we have y = x^3, and we say dy/dx = 3x^2.
If we looked at y^2, it would be said to be x^6, so y^2 = x^6.

Taking the derivative of the left side has always been done with just a y,
but it works with more than that.  In this case, 2y(x)(dy/dx) = 6x^5.
Since we know y = x^3, we known that 2y = 2x^3.  With this, the left side
can be divided by 2y and the right side by 2x^3.  The result is
dy/dx = 3x^2, which is what we got originally.

This problem
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We had A(r), and wanted to find dA/dt.
We know that A(r) = 4*pi*r^2, and here we have r(t) is found to be 3/80 when r is 10.
To express the problem in terms of time, since A is a function of r, and r is a function of t,
we have A(t) = 4*pi*r^2(t).  Thus, dA/dt = 8*pi*r(t)*dr/dt.

Since r(t) = 10, all we need is dr/dt, which is gotten from dV/dt = (dV/dr)(dr/dt)
in a similar fashion.