Question

We were given this statement about Rates of Change and told it is False but I don't understand why. "If we blow up a balloon so that the volume changes with a constant rate, then the radius (r(t)) changes with a constant rate." Thanks :)

Answer 1

v= 4/3 pi r^3
*derivative*
dv/dt = 4 pi r^2 dr/dt

Answer 2

but we weren't given the equation for V so how is it false for every function?

Answer 3

I'm assuming it's a sphere because I don't really know any other 3D shapes that are usually described with a radius

Answer 4

Im still a bit confused...sorry...how does taking the deriv. of V with respect to t show us that? Sorry..

Answer 5

I have...but I just can't see what your saying. to find the ROC of the volume wouldn't you do deriv wrt r. Why t?

Answer 6

oh :)

Answer 7

So are you saying...
for V' to be a constant (V'=4pir^2)
then r(t) needs to be a constant.
and if r(t) is a constant then r'(t) will be = 0

Answer 8

dv/dt = constant; that is one of the givens

dv/dt = 4 pi r^2 dr/dt

dv/dt being constant doesn't NECESSARILY imply that dr/dt is constant. You can find counterexamples for which a having a constant dv/dt doesn't entail any specific dr/dt because there is also a r^2 term

Answer 9

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

(var1) = 4 pi *(var2)^2 * (var3)

there are three variables in the equation

so even if you know one variable (dv/dt), you can't say one of the other variables (dr/dt) is constant because the equation is also dependent on a third variable (r)

Answer 10

oh ok thanks :)